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                 ‫5‬

                                                       ‫اﻟﻜﻔﺎءات اﻟﻤﺴﺘﮭﺪﻓﺔ‬
                                              ‫ﺣﺴﺎب ﻧﮭﺎﯾﺔ داﻟﺔ ﻋﻨﺪﻣﺎ ﯾﺆول ‪ x‬إﻟﻰ 0 ‪ x‬أو‬

‫‪ v‬ﺑﻌ ﺪ ﺗﻘ ﺪﯾﻢ دراﺳ ﺔ ﻧﮭﺎﯾ ﺎت داﻟ ﺔ ﻋﻨ ﺪ‬                          ‫إﻟﻰ ∞+ أو إﻟﻰ ∞− .‬

‫أﻃ ﺮاف ﻣﺠﻤﻮﻋ ﺔ ﺗﻌﺮﯾﻔﮭ ﺎ ﯾﺒﻘ ﻰ‬                 ‫ﻣﻌﺮﻓﺔ ﻧﮭﺎﯾﺔ داﻟﺔ ﻋﻨﺪﻣﺎ ﯾﺆول ‪ x‬إﻟﻰ 0 ‪ x‬أو‬

‫اﻟﮭﺪف اﻷﺳﺎﺳﻲ ﻣﻦ ھﺬا اﻟﻔﺼﻞ إﺗﻤﺎم‬                                  ‫إﻟﻰ ∞+ أو إﻟﻰ ∞− .‬
                                              ‫ﺣﺴﺎب ﻧﮭﺎﯾﺔ داﻟﺔ ﻧﺎﻃﻘﺔ ﻋﻨﺪ ﻋﺪد ‪ a‬ﺣﯿﺚ ‪a‬‬
‫ﺗﻜﻮﯾﻦ اﻟﻤﺘﻌﻠﻢ و ﺟﻌﻠﮫ أﻛﺜﺮ اﺳ ﺘﻘﻼﻟﯿﺔ‬
                                                        ‫ﺣﺪ ﻟﻤﺠﻤﻮﻋﺔ ﺗﻌﺮﯾﻒ ھﺬه اﻟﺪاﻟﺔ.‬
‫ﻓﯿﻤ ﺎ ﯾﺨ ﺺ اﻟﺪراﺳ ﺔ اﻟﺘﺎﻣ ﺔ ﻟﺪاﻟ ﺔ‬
                                                  ‫اﻟﺘﻔﺴﯿﺮ اﻟﺒﯿﺎﻧﻲ ﻟﻨﮭﺎﯾﺔ ﻏﯿﺮ ﻣﻨﺘﮭﯿﺔ ﻟﺪاﻟﺔ‬
‫اﻧﻄﻼﻗ ﺎ ﻣ ﻦ ﻋﺒﺎرﺗﮭ ﺎ اﻟﺠﺒﺮﯾ ﺔ ﺛ ﻢ‬                               ‫ﻋﻨﺪﻣﺎ ﯾﺆول ‪ x‬إﻟﻰ 0 ‪. x‬‬
‫ﺗﻤﺜﯿﻠﮭﺎ ﺑﯿﺎﻧﯿﺎ و ذﻟﻚ ﻣﻦ ﺧ ﻼل دراﺳ ﺔ‬            ‫ﻣﻌﺮﻓﺔ ﺷﺮط وﺟﻮد ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﯾﻮازي‬
‫اﻟﺪوال اﻟﻤﻨﺼﻮص ﻋﻠﯿﮭﺎ ﻓﻲ اﻟﺒﺮﻧﺎﻣﺞ‬                                    ‫أﺣﺪ ﻣﺤﻮري اﻟﻤﻌﻠﻢ.‬
‫و ھ ﻲ اﻟ ﺪوال ﻛﺜﯿ ﺮات اﻟﺤ ﺪود و‬                     ‫ﺗﺒﺮﯾﺮ أن ﻣﺴﺘﻘﯿﻤﺎ ﻣﻌﻠﻮﻣﺎ ھﻮ ﻣﺴﺘﻘﯿﻢ‬
              ‫اﻟﺪوال اﻟﻨﺎﻃﻘﺔ اﻟﺒﺴﯿﻄﺔ.‬            ‫ﻣﻘﺎرب.اﻟﺒﺤﺚ ﻋﻦ ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﻣﺎﺋﻞ.‬

‫‪ v‬ﯾ ﺘﻢ ﻛ ﺬﻟﻚ ﻓ ﻲ ھ ﺬا اﻟﻔ ﺼﻞ دراﺳ ﺔ‬               ‫اﺳﺘﻌﻤﺎل اﻟﻨﻈﺮﯾﺎت اﻷوﻟﯿﺔ) اﻟﻤﺠﻤﻮع،‬
                                              ‫اﻟﺠﺪاء، اﻟﻤﻘﻠﻮب و ﺣﺎﺻﻞ اﻟﻘﺴﻤﺔ( ﻟﺤﺴﺎب‬
‫اﻟ ﺴﻠﻮك اﻟﺘﻘ ﺎرﺑﻲ ﻟﻤﻨﺤﻨ ﻲ داﻟ ﺔ ﻣ ﻦ‬
                                                                                 ‫ﻧﮭﺎﯾﺎت.‬
‫ﺧﻼل ﺗﻌﯿﯿﻦ اﻟﻤﺴﺘﻘﯿﻤﺎت اﻟﻤﻘﺎرﺑﺔ ﻟﮫ )‬
                                                     ‫ﺣﺴﺎب ﻧﮭﺎﯾﺎت ﺑﺈزاﻟﺔ ﻋﺪم اﻟﺘﻌﯿﯿﻦ.‬
‫إن وﺟ ﺪت ( و اﻟﻤﻮازﯾ ﺔ ﻟﻤﺤ ﻮر‬
‫اﻟﻔﻮاﺻﻞ أو ﻣﺤ ﻮر اﻟﺘﺮاﺗﯿ ﺐ اﻧﻄﻼﻗ ﺎ‬
‫ﻣ ﻦ ﺣ ﺴﺎب اﻟﻨﮭﺎﯾ ﺎت و ﻛ ﺬﻟﻚ ﺗﻌﯿ ﯿﻦ‬
‫اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﻘﺎرب اﻟﻤﺎﺋ ﻞ ) إن وﺟ ﺪ (‬
‫إﻣ ﺎ ﺑﻌﻤﻠﯿ ﺔ اﻟﺒﺤ ﺚ ﻋﻠﯿ ﮫ أو اﺳ ﺘﻨﺘﺎﺟﮫ‬
    ‫اﻧﻄﻼﻗﺎ ﻣﻦ اﻟﻌﺒﺎرة اﻟﺠﺒﺮﯾﺔ ﻟﻠﺪاﻟﺔ.‬


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                                                                                          ‫اﻷﻧﺸﻄﺔ‬
                                                ‫اﻟﻨﺸﺎط 3 :‬                                                    ‫اﻟﻨﺸﺎط 1 :‬
                       ‫اﻟﮭﺪف :ﻧﮭﺎﯾﺔ ﻏﯿﺮ ﻣﻨﺘﮭﯿﺔ ﻋﻨﺪ ﻣﺎﻻﻧﮭﺎﯾﺔ.‬                   ‫اﻟﮭﺪف : ﻧﮭﺎﯾﺔ ﻏﯿﺮ ﻣﻨﺘﮭﯿﺔ ﻟﺪاﻟﺔ ﻋﻨﺪ ﻋﺪد.‬
                                                                                                                   ‫1(‬
    ‫3( ﻧﻼﺣﻆ أن ) ‪ k ( x‬ﺗﺄﺧﺬ ﻗﯿﻤﺎ ﻛﺒﯿﺮة ﺟﺪا أﻛﺜﺮ ﻓﺄﻛﺜﺮ ﻛﻠﻤﺎ‬
                                                                   ‫‪x‬‬        ‫∞−‬               ‫3‬                  ‫∞+‬
                                        ‫اﻗﺘﺮب ‪ x‬ﻣﻦ اﻟﻌﺪد 1.‬     ‫) ‪f ′(x‬‬           ‫+‬                      ‫-‬
     ‫4( ‪ x 2 ≥ A‬ﯾﻌﻨﻲ ‪ x ≤ − A‬أو ‪ . x ≥ A‬و ﺑﺎﻟﺘﺎﻟﻲ‬
                                        ‫ﯾﻜﻔﻲ أﺧﺬ ‪. B = A‬‬
                                                                 ‫) ‪f (x‬‬
                                                ‫اﻟﻨﺸﺎط 4 :‬
                          ‫اﻟﮭﺪف : ﻧﮭﺎﯾﺔ ﻣﻨﺘﮭﯿﺔ ﻋﻨﺪ ﻣﺎﻻﻧﮭﺎﯾﺔ.‬                                                         ‫2(‬
                                        ‫1( 2 = ‪ a‬و 1 = ‪. b‬‬        ‫‪x‬‬         ‫9.2‬            ‫99.2‬      ‫999.2‬     ‫9999.2‬
                                                         ‫2(‬     ‫) ‪f (x‬‬      ‫201‬             ‫401‬       ‫601‬        ‫801‬
  ‫‪x‬‬            ‫∞−‬                    ‫0‬                ‫∞+‬          ‫‪x‬‬       ‫1000.3‬           ‫100.3‬      ‫10.3‬       ‫1.3‬
‫) ‪h′ (x‬‬                   ‫-‬                     ‫-‬               ‫) ‪f (x‬‬      ‫801‬             ‫601‬       ‫401‬        ‫201‬
                                                                 ‫3( ﻛﻠﻤﺎ اﻗﺘﺮب ‪ x‬ﻣﻦ 3 إﻻ و أﺧﺬ ) ‪ f ( x‬ﻗﯿﻤﺎ ﻛﺒﯿﺮة ﺟﺪا.‬
    ‫) ‪h (x‬‬
                                                                                  ‫4( إذا أﺧﺬﻧﺎ ﻣﺜﻼ 4−01 + 3 ≤ ‪ 3 < x‬ﻓﺈن‬
                                                                              ‫1‬
                                                                                         ‫8− 01 ≤ )3 − ‪ 0 < ( x‬و ﻣﻨﮫ 801 ≥‬
                                                                                                                   ‫2‬
                                                           ‫3(‬
                                                                          ‫)3 − ‪( x‬‬
                                                                                     ‫2‬

  ‫‪x‬‬          ‫01-‬       ‫301−‬    ‫501−‬               ‫701−‬
‫) ‪h (x‬‬        ‫9.1‬     ‫999999.1 999.1‬              ‫...9.1‬            ‫− 3 ﻣﻊ 3 ≠ ‪ x‬ﻓﺈن‬
                                                                                        ‫1‬
                                                                                           ‫+3 ≤ ‪≤ x‬‬
                                                                                                      ‫1‬
                                                                                                          ‫5( إذا ﻛﺎن‬
                                                                                        ‫‪A‬‬              ‫‪A‬‬
  ‫‪x‬‬           ‫01‬       ‫301‬           ‫501‬          ‫701‬
                                                                                          ‫1‬                     ‫1‬
‫) ‪h (x‬‬                                                          ‫≤ )3 − ‪ 0 < ( x‬و ﺑﺎﻟﺘﺎﻟﻲ‬     ‫≤ 3 − ‪ 0 < x‬وﻣﻨﮫ‬
                                                                                      ‫2‬
              ‫1.2‬     ‫100.2‬        ‫10000.2‬       ‫...00.2‬
                                                                                          ‫‪A‬‬                      ‫‪A‬‬
       ‫4( ﻧﻼﺣﻆ أﻧﮫ ﻛﻠﻤﺎ أﺧﺬت ‪ x‬ﻗﯿﻤﺎ ﻛﺒﯿﺮة أﻛﺜﺮ ﻓﺄﻛﺜﺮ ﻓﺈن‬                                   ‫‪f (x ) ≥ A‬‬
                                   ‫) ‪ h ( x‬ﺗﻘﺘﺮب ﻣﻦ اﻟﻌﺪد2.‬                                                   ‫اﻟﻨﺸﺎط 2 :‬
                    ‫1‬                                           ‫اﻟﮭﺪف : ﻧﮭﺎﯾﺔ ﻏﯿﺮ ﻣﻨﺘﮭﯿﺔ ﻟﺪاﻟﺔ ﻋﻨﺪ ﻋﺪد ﻣﻦ اﻟﯿﻤﯿﻦ)اﻟﯿﺴﺎر(.‬
                      ‫5( ﺑﻔﺮض 601 ≥ ‪ x‬ﯾﻜﻮن 6−01 ≤‬
         ‫< 0 و ﺑﺎﻟﺘﺎﻟﻲ:‬
                    ‫‪x‬‬                                                                                                 ‫1(‬
                                      ‫1‬                             ‫‪x‬‬         ‫∞−‬                ‫1‬                  ‫∞+‬
                             ‫6−01 + 2 ≤ + 2 < 2 .‬                ‫‪g‬‬ ‫) ‪′ (x‬‬            ‫-‬
                                      ‫‪x‬‬                                                                     ‫-‬
           ‫1‬           ‫1‬
      ‫6( ‪ 2 < h ( x ) ≤ 2 + e‬ﯾﻌﻨﻲ ≥ ‪ . x‬ﻧﺄﺧﺬ ≥ ‪. B‬‬
           ‫‪e‬‬           ‫‪e‬‬                                         ‫) ‪g (x‬‬

                                                ‫اﻟﻨﺸﺎط 5 :‬                                                       ‫2(‬
                              ‫اﻟﮭﺪف : ﻧﮭﺎﯾﺔ ﻣﻨﺘﮭﯿﺔ ﻋﻨﺪ ﻋﺪد.‬       ‫‪x‬‬         ‫9.0‬            ‫99.0‬      ‫9999.0 999.0‬
                                                        ‫1(‬      ‫) ‪g (x‬‬      ‫01-‬            ‫001-‬      ‫0001-‬  ‫401−‬
  ‫‪x‬‬             ‫799.1‬            ‫899.1‬         ‫999.1‬
                                                                          ‫1000.1‬           ‫100.1‬      ‫10.1‬   ‫1.1‬
‫) ‪f (x‬‬                                                            ‫‪x‬‬
                ‫799.2‬            ‫899.2‬         ‫999.2‬
                                                                ‫) ‪g (x‬‬      ‫401‬            ‫0001‬       ‫001‬    ‫01‬
  ‫‪x‬‬             ‫100.2‬            ‫200.2‬         ‫300.2‬
‫) ‪f (x‬‬          ‫100.3‬            ‫200.3‬         ‫300.3‬            ‫3( ﻛﻠﻤﺎ اﻗﺘﺮب ‪ x‬ﻣﻦ 1 ﻓﺈن ) ‪ f ( x‬ﺗﺄﺧﺬ ﻗﯿﻤﺎ ﻛﺒﯿﺮة أﻛﺜﺮ‬

‫2( ﻧﻼﺣﻆ: ﻛﻠﻤﺎ اﻗﺘﺮب ‪ x‬ﻣﻦ 2 إﻻ و اﻗﺘﺮب ) ‪ f ( x‬ﻣﻦ 3‬                                                             ‫ﻓﺄﻛﺜﺮ.‬
                                                                                  ‫01−‬                        ‫01−‬
                                                                  ‫4( ﺑﻔﺮض 01 + 1 ≤ ‪ 1 < x‬ﯾﻜﻮن 01 ≤ 1 − ‪0 < x‬‬
                          ‫3( ﻣﻦ أﺟﻞ 2 ≠ ‪f ( x ) = x + 1 ، x‬‬
                                                                                               ‫و ﻣﻨﮫ 0101 ≥ ) ‪. g (x‬‬
‫4( ‪ 0 ≤ f ( x ) − 3 < e‬ﯾﻌﻨﻲ ‪ 0 ≤ x − 2 < e‬و ﺑﺎﻟﺘﺎﻟﻲ‬                        ‫ـ‬
                                                                      ‫5( ﯾﻜﻔﻲ ﺗﻌﻮﯾﺾ، ﻓﻲ اﻟﺒﺮھﺎن اﻟﺴﺎﺑﻖ، 0101 ﺑِ ‪. A‬‬
                                           ‫ﯾﻜﻔﻲ أﺧﺬ ‪. α ≤ e‬‬

‫2‬
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                                     ‫دراﺳﺔ داﻟﺔ ﺗﻨﺎﻇﺮﯾﺔ:‬
                                                                             ‫اﻷﻋﻤﺎل اﻟﻤﻮﺟﮭﺔ‬
       ‫اﻟﮭﺪف: اﻟﺘﻌﺮف ﻋﻠﻰ ﻣﻨﺤﻨﻲ داﻟﺔ ﺗﻨﺎﻇﺮﯾﺔ و ﺧﻮاﺻﮫ‬                      ‫دراﺳﺔ داﻟﺔ ﻛﺜﯿﺮ ﺣﺪود ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ:‬
     ‫اﻟﺘﻌﺮﯾﻒ: إذا ﻛﺎن 0 = ‪ c‬و 0 ≠ ‪ d‬ﻓﺈن ‪ f‬داﻟﺔ ﺗﺂﻟﻔﯿﺔ.‬         ‫اﻟﮭﺪف: اﻟﺘﻌﺮف ﻋﻠﻰ ﺧﻮاص داﻟﺔ ﻛﺜﯿﺮ ﺣﺪود ﻣﻦ اﻟﺪرﺟﺔ3‬
                   ‫إذا ﻛﺎن 0 = ‪ ad − bc‬ﻓﺈن ‪ f‬داﻟﺔ ﺛﺎﺑﺘﺔ.‬         ‫اﻟﻤﺜﺎل: ) 3 ‪lim f ( x ) = lim x 3 (1 − 3 / x 2 + 1/ x‬‬
                                                                  ‫∞→ ‪x‬‬               ‫∞→ ‪x‬‬
                               ‫‪‬‬      ‫‪d  d‬‬           ‫‪‬‬                   ‫∞−‬               ‫1−‬          ‫1+‬                ‫∞+‬
                       ‫‪D f =  −∞, −  U  − , +∞ ‬‬              ‫‪x‬‬
                               ‫‪‬‬      ‫‪c  c‬‬           ‫‪‬‬      ‫‪f‬‬ ‫) ‪′(x‬‬             ‫+‬         ‫0‬         ‫0 -‬             ‫+‬
                                                  ‫اﻟﻤﺜﺎل:‬
                                                                                             ‫3‬                            ‫∞+‬
                                 ‫[∞+ ,1] ‪D f = ]−∞,1[ U‬‬       ‫) ‪f (x‬‬
                                         ‫2 = ‪ a‬و 1= ‪b‬‬                      ‫∞−‬                             ‫1−‬
   ‫‪x‬‬           ‫∞−‬                ‫1‬                ‫∞+‬
‫) ‪f ′(x‬‬                ‫-‬                                                                               ‫1 + ‪( ∆ ) : y = −3x‬‬
                                           ‫-‬
             ‫2‬                       ‫∞+‬
                                                                                            ‫‪f‬‬
                                                                                            ‫‪‬‬    ‫3 ‪( x ) − ( −3x + 1) = x‬‬
                                                                                                                    ‫‪‬‬
‫) ‪f (x‬‬                                                            ‫‪x‬‬             ‫1−‬                    ‫0‬                   ‫1+‬
                           ‫∞−‬                        ‫2‬        ‫‪f (x ) − y‬‬                    ‫-‬         ‫0‬           ‫+‬
                    ‫اﻟﻤﺴﺘﻘﯿﻤﺎن اﻟﻤﻘﺎرﺑﺎن: 1 = ‪ x‬و 2 = ‪y‬‬                                     ‫‪y‬‬
                 ‫1− = ) ‪ f ′ ( x‬ﯾﻌﻨﻲ 0 = ‪ x‬أو 2 = ‪. x‬‬                                       ‫3‬
                   ‫‪y‬‬
                   ‫4‬                                                                        ‫2‬


                   ‫3‬                                                                        ‫1‬


                   ‫2‬
                                                                    ‫2-‬          ‫1-‬          ‫0‬             ‫1‬           ‫‪2 x‬‬

                   ‫1‬                                                                    ‫1-‬


     ‫2-‬     ‫1-‬      ‫0‬        ‫1‬        ‫2‬       ‫3‬       ‫‪x‬‬                        ‫ﻗﻮاﻋﺪ ﺗﻐﯿﯿﺮ اﻟﻤﻌﻠﻢ: ‪ x = X‬و 1 + ‪y = Y‬‬
      ‫ﻗﻮاﻋﺪ ﺗﻐﯿﯿﺮ اﻟﻤﻌﻠﻢ: 1 + ‪ x = X‬و 2 + ‪y = Y‬‬                             ‫اﻟﻨﻘﻄﺔ )1,0 ( ‪ Ω‬ﻣﺮﻛﺰ ﺗﻨﺎﻇﺮ ﻟﻠﻤﻨﺤﻨﻲ ) ‪. (C f‬‬
            ‫‪r u‬‬ ‫‪r‬‬                                                                         ‫ﻧﺒﯿﻦ أن 0 < ) 4.0 ( ‪f ( 0.3) × f‬‬
‫= ‪Y‬‬
    ‫1‬
    ‫‪X‬‬
             ‫(‬          ‫)‬
      ‫ﻣﻌﺎدﻟﺔ ) ‪ (C f‬ﺑﺎﻟﻨﺴﺒﺔ إﻟﻰ اﻟﻤﻌﻠﻢ ‪ Ω; i , j‬ھﻲ‬
                                                                                 ‫3.0 و 4.0 ھﻤﺎ ﻗﯿﻤﺘﺎن ﻣﻘﺮﺑﺘﺎن ﻟﻠﻌﺪد ‪. α‬‬
                                          ‫اﻟﺘﻄﺒﯿﻖ:‬                                ‫ﻟﺪﯾﻨﺎ: 0 < ) 6.1( ‪ f (1.5 ) × f‬و ﺑﺎﻟﺘﺎﻟﻲ‬
                                 ‫‪y‬‬
                                                              ‫6.1 < ‪ 1.5 . 1.5 < β‬ﻗﯿﻤﺔ ﻣﻘﺮﺑﺔ إﻟﻰ1.0 ﺑﺎﻟﻨﻘﺼﺎن ﻟِ ‪. β‬‬
                                                                  ‫ـ‬
                                 ‫3‬
                                                                            ‫اﻟﺘﻄﺒﯿﻖ: ﻓﺎﺻﻠﺘﺎ ﻧﻘﻄﺘﻲ اﻟﺘﻘﺎﻃﻊ ھﻤﺎ 2- و 1.‬
                                                                                                   ‫1‬
                                                                                                              ‫‪y‬‬
                                 ‫2‬
                                                                                     ‫2-‬          ‫1-‬           ‫0‬           ‫1‬    ‫‪x‬‬
                                 ‫1‬
                                                                                                          ‫1-‬

     ‫4-‬    ‫3-‬     ‫2-‬    ‫1-‬       ‫0‬        ‫1‬       ‫2‬   ‫‪3x‬‬                                                  ‫2-‬
                             ‫1-‬
                                                                                                          ‫3-‬

                             ‫2-‬
                                                                                                          ‫4-‬

                             ‫3-‬
                   ‫)5.0 ;5.0− (‬      ‫ﻣﺮﻛﺰ اﻟﺘﻨﺎﻇﺮ ھﻲ اﻟﻨﻘﻄﺔ‬                            ‫) 2− ,1− (‬   ‫ﻣﺮﻛﺰ اﻟﺘﻨﺎﻇﺮ ھﻲ اﻟﻨﻘﻄﺔ‬


‫3‬
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         + ∞ (6            - ∞ (5
                                              3
                                                (4                         ‫ﺗــﻤـــﺎرﯾــــﻦ‬
                                             3                                                                  .‫1 ﺻﺤﯿﺢ‬
                 . 3 (2                   . 0 (1       17
                 .+ ∞ (4                  . 3 (3                                                                .‫2 ﺻﺤﯿﺢ‬

                                          1      18                                                             .‫ﺻﺤﯿﺢ‬
                                           − (1                                                                           3
                                          3
                                                                                                                 .‫4 ﺧﻄﺄ‬
       lim 1 f ( x ) = +∞ ، lim 1 f ( x ) = −∞ (2
     x →                     x →
        > 2                       < 2
                                                                                                                 .‫5 ﺧﻄﺄ‬
       lim f ( x ) = +∞ ،        lim f ( x ) = −∞ (3
     x → 0                   x → 0
        <                         >
                                                                                                                     (3 6
                                          .+ ∞ (4
      lim f ( x ) = +∞ ،         lim f ( x ) = −∞ (5                                                                 (2 7
    x → −2                  x → −2
       <                         >

       lim f ( x ) = +∞ ، lim f ( x ) = −∞ (6                                                                        (3 8
      x → 1                   x → 1
         >                         <

                                                                                                                     (3   9
        lim f ( x ) = +∞ , lim f ( x ) = +∞ (1 19
       x → −∞                  x → +∞

        lim f ( x ) = +∞ , lim f ( x ) = −∞ (2                   Df = ℜ                                                   10
       x → −∞                  x → +∞
                                                                     lim f ( x ) = −∞ , lim f ( x ) = 1
        lim f ( x ) = −∞ , lim f ( x ) = +∞ (3                   x → −∞                            x → +∞
       x → −∞                  x → +∞
                                                                      ∗
        lim f ( x ) = +∞ , lim f ( x ) = +∞ (4              Df = ℜ                                                        11
       x → −∞                  x → +∞
                                                            lim f ( x ) = 2 , lim f ( x ) = 2
        lim f ( x ) = −∞ , lim f ( x ) = −∞ (5              x → −∞                      x → +∞
       x → −∞                  x → +∞
                                                              lim f ( x ) = −∞ , lim f ( x ) = +∞
        lim f ( x ) = −∞ , lim f ( x ) = +∞ (6              x → 0
                                                               <
                                                                                           x → 0
                                                                                              >
       x → −∞                  x → +∞
                                                                                                                           .
                lim f ( x ) = 0 , lim f ( x ) = 0 (1 20      Df = [− 2 ,1[ ∪ ] , + ∞[
                                                                             1                                            12
               x → −∞            x → +∞
                                                             f ( −2 ) = − 1 , lim f ( x ) = 2
        lim f ( x ) = +∞ , lim f ( x ) = +∞ (2                                      x → +∞
       x → −∞                  x → +∞
                                                               lim f ( x ) = +∞ , lim f ( x ) = −∞
                lim f ( x ) = 0 , lim f ( x ) = 0 (3         x → 1
                                                                <
                                                                                             x → 1
                                                                                                >
               x → −∞            x → +∞

        lim f ( x ) = −2 , lim f ( x ) = −2      (4                                         :‫31 ﺗﺼﺤﯿﺢ‬
        x → −∞                x → +∞                           lim k ( x ) = +∞ ‫ و‬lim g( x ) = −∞
                                                            x → −1                             x → +∞
                                                               <
       lim g ( x ) = −∞ , lim f ( x ) = +∞             21                         .h    ‫اﻟﻤﻨﺤﻨﻰ اﻷول ﯾﻤﺜﻞ اﻟﺪاﻟﺔ‬
      x → +∞                  x → +∞
                                                 (1                                .k   ‫اﻟﻤﻨﺤﻨﻰ اﻟﺜﺎﻧﻲ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ‬
       lim ( f ( x ) + g( x )) = +∞                                               .g    ‫اﻟﻤﻨﺤﻨﻰ اﻟﺜﺎﻟﺚ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ‬
      x → +∞

       lim g ( x ) = −∞ , lim f ( x ) = +∞                                         .f   ‫اﻟﻤﻨﺤﻨﻰ اﻟﺮاﺑﻊ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ‬
      x → +∞                  x → +∞
                                                 (2
       lim ( f ( x ) + g( x )) = 3                                                                            1           14
      x → +∞
                                                                           1 (2                             −   (1
       lim g ( x ) = −∞ , lim f ( x ) = +∞                                                                    5
      x → +∞                  x → +∞
                                                 (3                       .- ∞ (4                           + ∞ (3
       lim ( f ( x ) + g( x )) = −∞
      x → +∞

       lim g ( x ) = −∞ , lim f ( x ) = +∞                                  .9 (2                             9 (1        15
      x → +∞                  x → +∞
                                                 (4                       .+ ∞ (4                           + ∞ (3
       lim ( f ( x ) + g( x )) = −∞
      x → +∞
                                                                            .1 (3            0 (2               0 (1      16


4
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          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬            ‫∞+ = ) ‪lim g( x ) = 0 , lim f ( x‬‬    ‫22‬
                                                          ‫∞+→ ‪. x‬‬                  ‫∞+ → ‪x‬‬
                                                                                               ‫1(‬
                             ‫52 1( ∞+ = ) ‪lim f ( x‬‬          ‫∞+ = )) ‪lim ( f ( x ) × g ( x‬‬
                                                           ‫∞+ → ‪x‬‬
                            ‫∞− → ‪x‬‬

      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬            ‫0 = ) ‪lim g( x ) = +∞ , lim f ( x‬‬
                                                            ‫∞+ → ‪x‬‬                  ‫∞+ → ‪x‬‬
                                                                                                   ‫2(‬
                            ‫2( ∞− = ) ‪lim f ( x‬‬             ‫0 = )) ‪lim ( f ( x ) × g ( x‬‬
                            ‫∞− → ‪x‬‬                          ‫∞+ → ‪x‬‬
      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬            ‫∞+ = ) ‪lim g( x ) = 0 , lim f ( x‬‬
                          ‫3( ﻻ ﯾﻤﻜﻦ ﺣﺴﺎب اﻟﻨﮭﺎﯾﺔ.‬           ‫∞+ → ‪x‬‬               ‫∞+ → ‪x‬‬
                                                                                                   ‫3(‬
                                ‫4( 1 = ) ‪lim f ( x‬‬          ‫2 = )) ‪lim ( f ( x ) × g ( x‬‬
                                  ‫∞− → ‪x‬‬                    ‫∞+ → ‪x‬‬

        ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                          ‫5( ﻻ ﯾﻤﻜﻦ ﺣﺴﺎب اﻟﻨﮭﺎﯾﺔ.‬
                              ‫6( ∞− = ) ‪lim f ( x‬‬          ‫∞+ = ) ‪lim g( x ) = 0 , lim f ( x‬‬            ‫32‬
                             ‫∞− → ‪x‬‬                        ‫∞+ → ‪x‬‬               ‫∞+ → ‪x‬‬

      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬                  ‫) ‪f( x‬‬                        ‫1(‬
                                                           ‫( ‪lim‬‬          ‫∞ += )‬
                             ‫7( 0 = ) ‪lim f ( x‬‬            ‫) ‪x → +∞ g ( x‬‬
                               ‫∞− → ‪x‬‬
        ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬              ‫0 = ) ‪lim g ( x ) = 0 , lim f ( x‬‬
                                                             ‫∞+ → ‪x‬‬                ‫∞+ → ‪x‬‬
                             ‫8( ∞− = ) ‪lim f ( x‬‬                    ‫)‪f(x‬‬                           ‫2(‬
                             ‫∞− → ‪x‬‬
                                                              ‫( ‪lim‬‬         ‫0= )‬
      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬             ‫) ‪x → +∞ g ( x‬‬

                             ‫9( ∞+ = ) ‪lim f ( x‬‬              ‫0 = ) ‪lim g ( x ) = 0 , lim f ( x‬‬
                              ‫∞− → ‪x‬‬                         ‫∞+ → ‪x‬‬                ‫∞+ → ‪x‬‬
      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬                    ‫)‪f(x‬‬                           ‫3(‬
                            ‫01( ∞− = ) ‪lim f ( x‬‬              ‫( ‪lim‬‬         ‫∞ += )‬
                             ‫∞− → ‪x‬‬                          ‫) ‪x → +∞ g ( x‬‬

      ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                                                                                     ‫42 1( 1 = ) ‪lim f ( x‬‬
                                                                                    ‫∞+ → ‪x‬‬
     ‫62 1( ∞− = ) ‪lim f ( x ) = +∞ , lim f ( x‬‬
    ‫0 →‪x ‬‬
       ‫>‬
                         ‫0 →‪x ‬‬
                            ‫<‬
                                                            ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬                           ‫2( ∞+ = ) ‪lim f ( x‬‬
                                                                                 ‫∞+ → ‪x‬‬
                            ‫2( ∞+ = ) ‪lim f ( x‬‬           ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                           ‫0→ ‪x‬‬

          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬                             ‫3( ∞− = ) ‪lim f ( x‬‬
                                                                                  ‫∞+ → ‪x‬‬
                            ‫3( ∞+ = ) ‪lim f ( x‬‬           ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                          ‫0 →‪x ‬‬
                             ‫>‬
                                                                               ‫4( 3− = ) ‪lim f ( x‬‬
          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬                                ‫∞+ → ‪x‬‬

                             ‫4( ∞− = ) ‪lim f ( x‬‬            ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                            ‫0→ ‪x‬‬
                                                                                ‫5( 0 = ) ‪lim f ( x‬‬
         ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬                                   ‫∞+ → ‪x‬‬

      ‫5( ∞+ = ) ‪lim f ( x ) = −∞ , lim f ( x‬‬                ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
     ‫0 →‪x ‬‬              ‫0 →‪x ‬‬
        ‫>‬                    ‫<‬                                                  ‫6( 3− = ) ‪lim f ( x‬‬
                                                                                   ‫∞+ → ‪x‬‬
          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                                                             ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                                ‫6( 0 = ) ‪lim f ( x‬‬
                                   ‫0→ ‪x‬‬                                          ‫7( 0 = ) ‪lim f ( x‬‬
                                                                                     ‫∞+ → ‪x‬‬
        ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                             ‫7( ∞+ = ) ‪lim f ( x‬‬             ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                           ‫0 →‪x ‬‬
                              ‫>‬
                                                                               ‫8( 2− = ) ‪lim f ( x‬‬
                                                                                  ‫∞+ → ‪x‬‬
          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬          ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
       ‫8( ∞+ = ) ‪lim f ( x ) = −∞ , lim f ( x‬‬                                 ‫9( ∞+ = ) ‪lim f ( x‬‬
     ‫0 →‪x ‬‬               ‫0 →‪x ‬‬                                                ‫∞+ → ‪x‬‬
        ‫>‬                     ‫<‬
         ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬         ‫ﻣﻨﺤﻨﻰ ‪ f‬ﻻ ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟـﻤﺤﻮر اﻟﻔﻮاﺻﻞ.‬
                                                                                ‫01( 4− = ) ‪lim f ( x‬‬
                                                                                   ‫∞+ → ‪x‬‬



‫5‬
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                  ‫‪y‬‬                                                         ‫72 1( 1 = ) ‪lim f ( x ) = 1 , lim f ( x‬‬
                  ‫6‬                                                        ‫∞− → ‪x‬‬             ‫∞+ → ‪x‬‬

                                                                     ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                  ‫5‬
                                                                          ‫2( 2 = ) ‪lim f ( x ) = 2 , lim f ( x‬‬
                                                                            ‫∞− → ‪x‬‬                 ‫∞+ → ‪x‬‬
                  ‫4‬
                                                                     ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                  ‫3‬                                                    ‫3( 2− = ) ‪lim f ( x ) = −2 , lim f ( x‬‬
                                                                          ‫∞− → ‪x‬‬                  ‫∞+ → ‪x‬‬
                  ‫2‬                                                  ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                  ‫1‬                                                       ‫4( 0 = ) ‪lim f ( x ) = 0 , lim f ( x‬‬
                                                                                 ‫∞− → ‪x‬‬               ‫∞+ → ‪x‬‬
                                                                     ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
                                                                          ‫5( 0 = ) ‪lim f ( x ) = 0 , lim f ( x‬‬
                      ‫0‬     ‫1‬       ‫2‬       ‫3‬   ‫4‬    ‫‪5 x‬‬
                                                                                 ‫∞− → ‪x‬‬               ‫∞+ → ‪x‬‬

                                                            ‫3(‬       ‫ﻣﻨﺤﻨﻰ ‪ f‬ﯾﻘﺒﻞ ﻣﻘﺎرب ﻣﻮازي ﻟﻤﺤﻮر اﻟﺘﺮاﺗﯿﺐ.‬
     ‫‪x‬‬       ‫‪-‬‬                         ‫1‬              ‫‪+‬‬

    ‫‪f'(x‬‬                                ‫0‬
       ‫)‬                                                            ‫‪x‬‬       ‫‪-‬‬                ‫1‬                 ‫‪+‬‬
                                                                                                                      ‫82 1(‬
                                        ‫4‬
                                                                  ‫)‪f'(x‬‬                       ‫0‬

    ‫)‪f(x‬‬                                                                    ‫‪+‬‬                                  ‫‪+‬‬

                                                                  ‫)‪f(x‬‬
             ‫‪-‬‬                                        ‫‪-‬‬
                           ‫‪y‬‬
                           ‫4‬                                                                  ‫0‬


                           ‫3‬                                                              ‫‪y‬‬
                           ‫2‬
                                                                                          ‫6‬

                           ‫1‬
                                                                                          ‫5‬
     ‫3-‬    ‫2-‬         ‫1-‬    ‫0‬       ‫1‬       ‫2‬   ‫3‬    ‫‪4 x‬‬
                           ‫1-‬                                                             ‫4‬

                           ‫2-‬
                                                                                          ‫3‬
                           ‫3-‬

                                                                                          ‫2‬

                                                     ‫∞-‬     ‫4(‬
           ‫∞- ‪x‬‬                         ‫2‬                                                 ‫1‬
           ‫('‪f‬‬                          ‫0‬
           ‫)‪x‬‬
                                        ‫2‬                                   ‫1-‬            ‫0‬   ‫1‬          ‫2‬     ‫‪3x‬‬
      ‫)‪f(x‬‬                                                                            ‫1-‬
                  ‫∞-‬                                 ‫∞-‬
                  ‫‪y‬‬
                  ‫2‬                                                 ‫‪x‬‬            ‫‪-‬‬               ‫5‬              ‫‪+‬‬
                                                                                                                      ‫2(‬
                                                                                                  ‫2‬

                  ‫1‬                                                ‫‪f'(x‬‬                           ‫0‬
                                                                      ‫)‬          ‫‪+‬‬                              ‫‪+‬‬
                  ‫0‬             ‫1‬       ‫2‬       ‫3‬     ‫‪4x‬‬
                                                                   ‫)‪f(x‬‬
             ‫1-‬

                                                                                                  ‫1-‬
             ‫2-‬                                                                                    ‫4‬

             ‫3-‬




‫6‬
                                                    ‫‪http://assil.yoo7.com‬‬
                                                                                                             ‫92 1(‬
             ‫)‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻤﯿﻦ ﻣﻘﺎرﺑﯿﻦ ﻣﻌﺎدﻟﺘﯿﮭﻤﺎ‬                                       ‫3‬               ‫3‬
                                  ‫0=‪ X‬و 0=‪.y‬‬                      ‫∞- ‪x‬‬              ‫−‬
                                                                                        ‫3‬               ‫3‬     ‫∞+‬
          ‫∞- ‪x‬‬                 ‫0‬                ‫4(‬              ‫)‪f'(x‬‬                   ‫0‬               ‫0‬
                                          ‫∞+‬
       ‫)‪f'(x‬‬                                                                        ‫763‬                       ‫∞+‬
                                                                                    ‫562‬
                          ‫∞+‬               ‫∞+‬                   ‫)‪f(x‬‬
                                                                                                    ‫361‬
         ‫)‪f(x‬‬                                                           ‫∞-‬                          ‫562‬
                   ‫-∞‬               ‫∞-‬                                   ‫أوﻻ ﻧﻐﯿﺮ رﻣﺰ اﻟﻨﻘﻄﺔ ﻟﯿﺼﺒﺢ ﻣﺜﻼ ‪ ω‬ﺛﻢ ﻧﺘﺒﻊ‬
                  ‫)‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻤﯿﻦ ﻣﻘﺎرﺑﯿﻦ ﻣﻌﺎدﻟﺘﯿﮭﻤﺎ‬                  ‫ﻃﺮﯾﻘﺔ ﺗﻐﯿﯿﺮ اﻟﻤﻌﻠﻢ:ﺑﺤﯿﺚ ﻧﻜﺘﺐ ﻣﻌﺎدﻟﺔ )‪(Cf‬‬
                                       ‫0=‪ X‬و 1=‪.y‬‬                                    ‫ﻓﻲ اﻟﻤﻌﻠﻢ )‪ (ω ;I ;J‬و ﺗﺼﺒﺢ:‬
                                                     ‫5(‬                 ‫‪ F(X)=X3-X‬ﺣﯿﺚ: 0+‪ X=x‬و 1+‪Y=y‬‬
        ‫‪x‬‬          ‫∞-‬                ‫5‬              ‫∞+‬                           ‫و ﻓﻲ اﻷﺧﯿﺮ ﻧﺜﺒﺖ أن ‪ F‬داﻟﺔ ﻓﺮدﯾﺔ‬
      ‫)‪f'(x‬‬                                                                                             ‫2(‬
                  ‫2‬                       ‫∞+‬                     ‫∞- ‪x‬‬             ‫2-‬                ‫2‬        ‫∞+‬
      ‫)‪f(x‬‬                                                     ‫‪f'(x‬‬                ‫0‬                ‫0‬
                                                                 ‫)‬                 ‫91‬
                                 ‫-∞‬                   ‫2‬                                                       ‫∞+‬
                                                                                   ‫3‬
                                                               ‫)‪f(x‬‬
                      ‫)‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻤﯿﻦ ﻣﻘﺎرﺑﯿﻦ ﻣﻌﺎدﻟﺘﯿﮭﻤﺎ‬                                         ‫31‬
                                                                                                 ‫−‬
                                         ‫5=‪ X‬و 2=‪.y‬‬                    ‫∞-‬                          ‫3‬
                                                       ‫6(‬               ‫إﺛﺒﺎت اﻟﻤﺮﻛﺰ ﯾﺘﻢ ﺑﻨﻔﺲ اﻟﻄﺮﯾﻘﺔ اﻟﺴﺎﺑﻘﺔ.‬
        ‫‪x‬‬   ‫∞-‬                        ‫2‬              ‫∞+‬                                                   ‫3(‬
      ‫)‪f'(x‬‬                                                       ‫∞- ‪x‬‬             ‫0‬                ‫2‬        ‫∞+‬
            ‫0‬                             ‫∞+‬                    ‫)‪f'(x‬‬                               ‫0‬
                                                                                   ‫0‬
      ‫)‪f(x‬‬                                                                                                    ‫∞+‬
                                                                                    ‫3‬
                                 ‫-∞‬                   ‫0‬        ‫)‪f(x‬‬
                      ‫)‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻤﯿﻦ ﻣﻘﺎرﺑﯿﻦ ﻣﻌﺎدﻟﺘﯿﮭﻤﺎ‬            ‫∞-‬                        ‫1-‬
                                 ‫2=‪ X‬و 0=‪.y‬‬                            ‫إﺛﺒﺎت اﻟﻤﺮﻛﺰ ﯾﺘﻢ ﺑﻨﻔﺲ اﻟﻄﺮﯾﻘﺔ اﻟﺴﺎﺑﻘﺔ.‬
                                                    ‫13 1(‬
         ‫‪x‬‬   ‫∞-‬                       ‫0‬              ‫∞+‬                                                      ‫03 1(‬
       ‫)‪f'(x‬‬                                                      ‫∞- ‪x‬‬                        ‫1-‬              ‫∞+‬
                                 ‫∞+‬                    ‫1‬        ‫)‪f'(x‬‬
       ‫)‪f(x‬‬                                                                              ‫∞+‬                      ‫2‬
                                                                 ‫)‪f(x‬‬
                   ‫1‬                      ‫-∞‬
                                              ‫ﺑﻤﺎأن:‬                      ‫2‬                        ‫∞-‬
    ‫0 = ) ‪ lim ( f ( x ) − y‬و 0 = ) ‪lim ( f ( x ) − y‬‬                 ‫)‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻤﯿﻦ ﻣﻘﺎرﺑﯿﻦ ﻣﻌﺎدﻟﺘﯿﮭﻤﺎ‬
    ‫∞+ → ‪x‬‬                            ‫∞− → ‪x‬‬
                   ‫ﻓﺈن: )∆( ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﻟﻤﻨﺤﻨﻰ اﻟﺪاﻟﺔ ‪.f‬‬                                ‫1-=‪ X‬و 2=‪.y‬‬
                                                     ‫2(‬                                                 ‫2(‬
      ‫∞- ‪x‬‬             ‫2 −‬            ‫0‬          ‫2‬    ‫∞+‬           ‫‪x‬‬     ‫∞-‬                ‫0‬                ‫∞+‬
    ‫)‪f'(x‬‬               ‫0‬                       ‫0‬                ‫)‪f'(x‬‬
                          ‫746‬          ‫∞+‬                                               ‫∞+ ∞+‬
                      ‫−‬                               ‫∞+‬
                          ‫961‬
     ‫)‪f(x‬‬                                                        ‫)‪f(x‬‬
                                               ‫373‬
             ‫-∞‬                  ‫∞-‬            ‫402‬                         ‫0‬                                     ‫0‬

‫7‬
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                     ‫‪x‬‬       ‫∞-‬                 ‫∞+‬        ‫53 1(‬                ‫ﻧﻔﺲ اﻟﻄﺮﯾﻘﺔ ﻹﺛﺒﺎت اﻟﻤﻘﺎرب اﻟﻤﺎﺋﻞ.‬
                  ‫)‪f'(x‬‬                                                                                       ‫3(‬
                                                                                   ‫961‬                         ‫804‬
                                                                     ‫∞- ‪x‬‬      ‫−‬                   ‫1‬
                                                ‫∞+‬                                 ‫804‬                         ‫961‬       ‫∞+‬
                  ‫)‪f(x‬‬                                             ‫)‪f'(x‬‬            ‫0‬                              ‫0‬
                                                                                   ‫961‬
                                                                               ‫−‬                       ‫∞+‬                 ‫∞+‬
                             ‫∞-‬                                                    ‫402‬
                                                                   ‫)‪f(x‬‬
     ‫ﺗﺴﺘﺨﺪم اﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ اﻟﺒﯿﺎﻧﯿﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ اﻟﻨﺘﺎﺋﺞ.‬                                                           ‫618‬
                                                  ‫2(‬                      ‫-∞‬                   ‫∞-‬              ‫961‬

       ‫33 − ∞- ‪x‬‬                              ‫3‬                                 ‫ﻧﻔﺲ اﻟﻄﺮﯾﻘﺔ ﻹﺛﺒﺎت اﻟﻤﻘﺎرب اﻟﻤﺎﺋﻞ.‬
                                 ‫0‬                ‫∞+‬  ‫3‬
                                                                                                              ‫4(‬
     ‫)‪f'(x‬‬               ‫0‬             ‫0‬             ‫0‬                             ‫932‬                         ‫775‬
             ‫∞+‬                                                      ‫∞- ‪x‬‬          ‫804‬              ‫2‬          ‫961‬        ‫∞+‬
                                                          ‫∞+‬       ‫)‪f'(x‬‬            ‫0‬                              ‫0‬
      ‫)‪f(x‬‬                             ‫3-‬            ‫21‬
                                                                                   ‫932‬                 ‫∞+‬                 ‫∞+‬
                      ‫82‬
                    ‫−‬                                                              ‫402‬
                      ‫9‬                                            ‫)‪f(x‬‬
                                                                                                               ‫4511‬
      ‫ﺗﺴﺘﺨﺪم اﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ اﻟﺒﯿﺎﻧﯿﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ اﻟﻨﺘﺎﺋﺞ.‬                     ‫-∞‬                   ‫∞-‬              ‫961‬
                                                   ‫3(‬
      ‫∞- ‪x‬‬          ‫0‬           ‫2‬           ‫4‬       ‫∞+‬                          ‫ﻧﻔﺲ اﻟﻄﺮﯾﻘﺔ ﻹﺛﺒﺎت اﻟﻤﻘﺎرب اﻟﻤﺎﺋﻞ.‬
    ‫)‪f'(x‬‬          ‫0‬            ‫0‬           ‫0‬
                                                                                                                              ‫23‬
           ‫∞+‬                   ‫7‬                  ‫∞+‬                                       ‫اﻟﻤﻨﺤﻨﻰ اﻷول ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.f‬‬
    ‫)‪f(x‬‬                                                                                   ‫اﻟﻤﻨﺤﻨﻰ اﻟﺜﺎﻧﻲ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.g‬‬
                                                                                           ‫اﻟﻤﻨﺤﻨﻰ اﻟﺜﺎﻟﺚ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.h‬‬
                  ‫9-‬                       ‫9-‬                                              ‫اﻟﻤﻨﺤﻨﻰ اﻟﺮاﺑﻊ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.k‬‬
     ‫ﺗﺴﺘﺨﺪم اﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ اﻟﺒﯿﺎﻧﯿﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ اﻟﻨﺘﺎﺋﺞ.‬                                     ‫اﻟﻤﻨﺤﻨﻰ اﻟﺨﺎﻣﺲ ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.l‬‬
                                                  ‫4(‬                                    ‫اﻟﻤﻨﺤﻨﻰ اﻟﺴﺎدس ﯾﻤﺜﻞ اﻟﺪاﻟﺔ ‪.m‬‬
       ‫∞- ‪x‬‬                  ‫1-‬                 ‫0‬         ‫∞+‬
                                                                                            ‫1‬                                 ‫33‬
     ‫)‪f'(x‬‬                    ‫0‬                  ‫0‬                                      ‫−‬     ‫3(‬        ‫2( 0‬           ‫1( 1‬
           ‫∞+‬                                                                               ‫2‬
                                                ‫1-‬
                                                                                                      ‫1‬
    ‫)‪f(x‬‬                       ‫3‬
                                                                                            ‫6( 21.‬      ‫4(‬  ‫5( 4‬
                             ‫−‬                                                                        ‫2‬
                                                   ‫-∞‬
                               ‫2‬
     ‫ﺗﺴﺘﺨﺪم اﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ اﻟﺒﯿﺎﻧﯿﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ اﻟﻨﺘﺎﺋﺞ.‬                ‫7( ∞− = ) ‪lim f ( x ) = +∞ , lim f ( x‬‬
                                                                   ‫3 →‪x ‬‬                     ‫3 →‪x ‬‬
                                                                      ‫>‬                           ‫<‬

                                                                                            ‫1‬
                                   ‫‪y‬‬                      ‫63 1(‬                                 ‫9(‬    ‫8( 3‬
                                                                                                        ‫.‬
                                                                                           ‫21‬
                                   ‫2‬                                ‫01( ∞+ = ) ‪lim f ( x ) = −∞ , lim f ( x‬‬
                                                                   ‫0 →‪x ‬‬                     ‫0 →‪x ‬‬
                                                                      ‫>‬                           ‫<‬

                                   ‫1‬                                ‫11( ∞− = ) ‪lim f ( x ) = +∞ , lim f ( x‬‬
                                                                   ‫1 →‪x ‬‬                     ‫1 →‪x ‬‬
                                                                      ‫>‬                           ‫<‬



                      ‫1-‬           ‫0‬        ‫1‬        ‫‪x‬‬                             ‫3( ∞ +‬           ‫2( ∞ -‬             ‫43 1( 0‬
                                                                                                      ‫3‬
                              ‫1-‬                                                        ‫6( 0‬        ‫5( −‬               ‫4( 0‬
                                                                                                      ‫4‬
                                                                                                                       ‫1‬
                              ‫2-‬                                                            ‫8( ∞ +.‬                      ‫7(‬
                                                                                                                       ‫2‬
                              ‫3-‬


‫8‬
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                                                                                                                         ‫‪y‬‬
                                                                                                                                                       ‫2(‬
                           ‫‪y‬‬                                                                                             ‫1‬
                           ‫2‬

                           ‫1‬

                                                                                                         ‫2-‬        ‫1-‬     ‫0‬              ‫1‬         ‫‪x‬‬
               ‫2-‬   ‫1-‬     ‫0‬            ‫1‬        ‫2‬        ‫3‬         ‫4‬       ‫5‬       ‫‪x‬‬
                          ‫1-‬                                                                                            ‫1-‬
                          ‫2-‬

                          ‫3-‬                                                                                            ‫2-‬
                          ‫4-‬

                          ‫5-‬
                                                                                                                        ‫3-‬
                          ‫6-‬

                                                                                                               ‫‪y‬‬

                                                                                                              ‫6‬
                                                                                                                                                       ‫3(‬
                                                                                    ‫2(‬                        ‫5‬


         ‫∞- ‪x‬‬                      ‫1-‬                           ‫1‬                   ‫∞+‬                        ‫4‬

                                                                                                              ‫3‬
       ‫)‪f'(x‬‬                           ‫0‬                       ‫0‬                                              ‫2‬

               ‫∞+‬                                              ‫1‬                                              ‫1‬


        ‫)‪f(x‬‬                                                                                             ‫1-‬    ‫0‬    ‫1‬    ‫2‬           ‫3‬   ‫4‬   ‫‪5 x‬‬

                                                                                                              ‫1-‬

                                   ‫1-‬                                               ‫∞-‬                        ‫2-‬

                                                                                                              ‫3-‬
                                             ‫‪y‬‬
                                             ‫1‬                                                                ‫4-‬

                                                                                                              ‫5-‬


         ‫5-‬    ‫4-‬   ‫3-‬     ‫2-‬          ‫1-‬    ‫0‬       ‫1‬          ‫2‬       ‫3‬       ‫4‬        ‫5‬   ‫‪x‬‬                ‫6-‬

                                                                                                              ‫7-‬
                                            ‫1-‬
                                                                                                              ‫8-‬


                                            ‫2-‬                                                                ‫9-‬




      ‫اﻷﺟﺰاء 3( 4( 5( 6( 7( ﯾﺘﻢ اﻹﺟﺎﺑﺔ ﻋﻠﯿﮭﺎ ﺑﻨﻔﺲ‬                                                                        ‫‪y‬‬                             ‫4(‬
                                          ‫اﻟﻄﺮﯾﻘﺔ.‬                                                                       ‫1‬

                    ‫83 1( ﻟﯿﻜﻦ ‪ x‬ﻋﺪد ﺣﻘﯿﻘﻲ ﻣﻦ ‪:D‬‬

         ‫= ) ‪f( x‬‬
                      ‫(‬
                   ‫‪x +3 − x x +3 + x‬‬                 ‫()‬                                 ‫)‬         ‫3-‬     ‫2-‬        ‫1-‬        ‫0‬           ‫1‬         ‫‪x‬‬

                           ‫‪x +3+ x‬‬     ‫(‬                                ‫)‬                                                ‫1-‬

                      ‫3‬
               ‫=‬                                                                                                         ‫2-‬
                  ‫‪x+3+ x‬‬
            ‫2( ﻟﺪﯾﻨﺎ ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪ x‬ﻣﻦ ‪:D‬‬                                                                         ‫3-‬
                                                              ‫)1( .......... ‪x‬‬
       ‫3‬
                    ‫0>‬         ‫و‬            ‫+3+‪x‬‬
     ‫+3+‪x‬‬       ‫‪x‬‬                                                                                                                                      ‫73 1(‬
     ‫+3+‪x‬‬       ‫> ‪x‬‬       ‫‪x‬‬        ‫و‬        ‫0> 3+‪x‬‬                                                ‫‪x‬‬      ‫∞-‬                  ‫3‬
                                                                                                                                               ‫∞+‬
                                                                                                                                 ‫2‬
                                ‫أي‬                                                               ‫)‪f'(x‬‬
                      ‫3‬                    ‫3‬
                                         ‫<‬    ‫) 2 (...... ..........‬                                                    ‫∞+‬                             ‫3‬
                    ‫+3+‪x‬‬               ‫‪x‬‬    ‫‪x‬‬
                                                                                                 ‫)‪f(x‬‬
    ‫ﻣﻦ )1( و )2( ﻧﺴﺘﻨﺘﺞ أﻧﮫ ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪x‬‬                                                        ‫3‬                       ‫∞-‬
                                       ‫3‬
                        ‫≤ ) ‪0 ≤ f( x‬‬       ‫ﻣﻦ ‪:D‬‬
                                        ‫‪x‬‬

‫9‬
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          ‫14 1( ﻟﺪﯾﻨﺎ ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪ x‬ﻣﻦ ‪: D‬‬                                                      ‫3‬
                                                                              ‫‪ lim‬ﻓﺈن:‬                     ‫3( ﺑﻤﺎأن: 0 =‬
                 ‫1+ ≤ ‪− 1 ≤ sin x‬‬                                                      ‫∞+ → ‪x‬‬
                                                                                                         ‫‪x‬‬
                       ‫1 + 2 ‪x 2 − 1 ≤ x 2 + sin x ≤ x‬‬                                                 ‫0 = ) ‪lim f ( x‬‬
                                                                                                    ‫∞+ → ‪x‬‬

                            ‫ﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ x‬ﻧﺠﺪ:‬                                                                                 ‫93‬
                                                                                                                          ‫1(‬
                      ‫1− ‪x‬‬    ‫2‬
                                           ‫1+ 2‪x‬‬
                              ‫≤ )‪≤ f(x‬‬                                     ‫0101 601 401 ‪x‬‬
                          ‫‪x‬‬                   ‫‪x‬‬                         ‫10,1 )‪f(x‬‬     ‫1‬     ‫1‬
                                         ‫2( ﺑﻤﺎأن:‬                         ‫01 ‪x‬‬ ‫21‬
                                                                                   ‫01‬ ‫02‬
                                                                                         ‫01‬ ‫04‬

                ‫1+ 2‪x‬‬                 ‫1− 2 ‪x‬‬                            ‫)‪f(x‬‬    ‫1‬     ‫1‬     ‫1‬
          ‫‪lim‬‬         ‫‪ lim‬و ∞+ =‬              ‫∞+ =‬
         ‫∞+ → ‪x‬‬   ‫‪x‬‬            ‫∞+ → ‪x‬‬    ‫‪x‬‬
                        ‫ﻓﺈن: ∞+ = ) ‪lim f ( x‬‬                   ‫2( ﻟﺪﯾﻨﺎ ﻣﻦ أﺟﻞ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ x‬اﻟﻤﻮﺟﺐ ﺗﻤﺎﻣﺎ:‬
                             ‫∞+ → ‪x‬‬

                                                              ‫1 + ‪ x 2 + x + 1 ≤ x 2 + 2x‬و 1 + ‪x 2 ≤ x 2 + x‬‬
                                                  ‫2‬
                                  ‫≤ ) ‪0 ≤ f( x‬‬        ‫24 1(‬                                            ‫أي‬
                                                  ‫‪x‬‬
                       ‫2‬                                                      ‫2 )1 + ‪x ≤ x + x + 1 ≤ ( x‬‬
                                                                               ‫2‬               ‫2‬

               ‫‪lim‬‬ ‫0 = ) ‪ lim f ( x‬ﻷن: 0 =‬                                                          ‫و ﻣﻨﮫ:‬
              ‫∞+ → ‪x‬‬          ‫∞+ → ‪x‬‬
                 ‫‪x‬‬
                                                                              ‫≤ ‪x‬‬              ‫‪x‬‬   ‫2‬
                                                                                                        ‫1 + ‪+ x + 1 ≤ x‬‬
         ‫ﻻ ﯾﻤﻜﻦ ﺣﺴﺎب اﻟﻨﮭﺎﯾﺔ ﻟﻤﺎ ‪ x‬ﯾﺆول إﻟﻰ ∞+.‬
                 ‫‪−x‬‬                    ‫‪x‬‬
                       ‫2 ≤ ) ‪≤ f( x‬‬          ‫2(‬
               ‫3+ ‪x‬‬
                 ‫2‬
                                     ‫3+ ‪x‬‬
                                                                ‫3( ﻟﺪﯾﻨﺎ ﻣﻦ أﺟﻞ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ x‬اﻟﻤﻮﺟﺐ ﺗﻤﺎﻣﺎ:‬
                            ‫0 = ) ‪ lim f ( x‬ﻷن:‬
                                       ‫∞+ → ‪x‬‬                                ‫≤ ‪x‬‬           ‫‪x‬‬   ‫2‬
                                                                                                   ‫1 + ‪+ x + 1 ≤ x‬‬
                     ‫‪−x‬‬                    ‫‪x‬‬
              ‫‪lim‬‬          ‫2 ‪= 0 , lim‬‬          ‫0=‬
              ‫∞+ → ‪x‬‬‫3+ ‪x‬‬
                     ‫2‬           ‫3 + ‪x → +∞ x‬‬                                              ‫ﺑﺤﺴﺎب ﻣﻘﻠﻮب اﻟﻌﺒﺎرة ﻧﺠﺪ.‬
                ‫ﺑﻨﻔﺲ اﻟﻄﺮﯾﻘﺔ ﯾﺘﻢ اﻹﺟﺎﺑﺔ ﻋﻠﻰ 3( و 4(.‬                 ‫1‬
                                                                         ‫≤‬
                                                                                                   ‫1‬
                                                                                                                 ‫≤‬
                                                                                                                     ‫1‬
                                                                   ‫1 + ‪x‬‬               ‫‪x‬‬   ‫2‬
                                                                                                   ‫1 + ‪+ x‬‬           ‫‪x‬‬

     ‫34 1( ﺑﻤﺎأن: 3 = ) ‪ lim f ( x‬و 3 = ) ‪lim f ( x‬‬                ‫ﺑﻀﺮب اﻟﻨﺘﯿﺠﺔ ﺑـ ‪ x + x‬ﻣﻊ اﻟﺘﺒﺴﯿﻂ ﻧﺠﺪ:‬
     ‫∞− → ‪x‬‬                  ‫∞+ → ‪x‬‬
                                                                          ‫1‬                                          ‫1‬
       ‫ﻓﺈن )‪ (Cf‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﻣﻌﺪﻟﺘﮫ 3=‪.y‬‬                     ‫− 1‬         ‫+ 1 ≤ ) ‪≤ f( x‬‬
                                                                        ‫1 + ‪x‬‬                                         ‫‪x‬‬
                 ‫2( ﺣﺴﺐ إﺷﺎرة اﻟﻔﺮق ‪:f(x) – y‬‬
     ‫ﻟﻤﺎ [∞ + , 1]∈‪ x‬ﻓﺈن )‪ (Cf‬ﯾﻘﻊ أﻋﻠﻰ )‪.(D‬‬                                                           ‫4( ﺑﻤﺎأن:‬
       ‫ﻟﻤﺎ [1 , ∞-]∈‪ x‬ﻓﺈن )‪ (Cf‬ﯾﻘﻊ أﺳﻔﻞ )‪.(D‬‬                                       ‫1‬                   ‫1‬
                                                                    ‫+ 1 ‪lim‬‬          ‫− 1 ‪ lim‬و 1 =‬         ‫1=‬
                                                                    ‫∞+ → ‪x‬‬
                                                                                   ‫‪x‬‬       ‫∞+ → ‪x‬‬   ‫‪1+ x‬‬
                         ‫44 1( 3=‪.a=-2 , b‬‬
         ‫إﺟﺎﺑﺔ اﻟﺴﺆاﻟﯿﻦ 2( و 3( ﻣﺜﻞ اﻟﺘﻤﺮﯾﻦ 34 .‬                                           ‫ﻓﺈن: 1 = ) ‪lim f ( x‬‬
                                                                                                        ‫∞+ → ‪x‬‬


                                         ‫54 1( ﺑﻤﺎأن:‬                    ‫04 ﻟﺪﯾﻨﺎ ﻣﻦ أﺟﻞ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ x‬ﻣﻦ ‪:D‬‬
              ‫0 = ‪ lim f ( x ) − y‬و 0 = ‪lim f ( x ) − y‬‬                                ‫1+ ≤ ‪− 1 ≤ sin x‬‬
          ‫∞− → ‪x‬‬                       ‫∞+ → ‪x‬‬

                       ‫ﻓﺈن: )‪ (C‬ﯾﻘﺒﻞ )∆( ﻛﻤﺴﺘﻘﯿﻢ ﻣﻘﺎرب.‬                                ‫4 ≤ ‪2 ≤ 3 + sin x‬‬
                        ‫2( دراﺳﺔ إﺷﺎرة اﻟﻔﺮق: ‪.f(x) – y‬‬                                 ‫ﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ x‬ﻧﺠﺪ:‬
                                                                                           ‫2‬               ‫4‬
                                                                                              ‫≤ ) ‪≤ f( x‬‬
                       ‫64 1( 71=‪a=2 , b=6 , c‬‬                                              ‫‪x‬‬               ‫‪x‬‬
                ‫ﻧﻔﺲ اﻟﻄﺮق اﻟﺴﺎﺑﻘﺔ ﻟﻺﺟﺎﺑﺔ ﻋﻠﻰ 2(‬                                                     ‫2( ﺑﻤﺎأن:‬
                                                                                     ‫2‬                ‫4‬
      ‫74 1( ( اﻟﺪاﻟﺔ ‪ h‬ھﻲ اﻟﺘﻲ ﺗﻮﻓﺮ اﻟﺸﺮوط اﻟﺴﺎﺑﻘﺔ.‬                            ‫0 = ‪ lim‬و 0 = ‪lim‬‬
                                                                              ‫‪x → +∞ x‬‬         ‫‪x → +∞ x‬‬
            ‫2( ﻻ ﯾﻤﻜﻦ ﺗﻌﯿﯿﻦ ﻗﯿﻤﺔ ‪ a‬ﻣﻦ أﺟﻞ 1=‪x‬‬
                                                                                                       ‫ﻓﺈن: 0 = ) ‪lim f ( x‬‬
                                                                                                   ‫∞+ → ‪x‬‬




‫01‬
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                                                                ‫( اﻟﺪاﻟﺔ ‪ k‬ھﻲ اﻟﺘﻲ ﺗﻤﺜﯿﻠﮭﺎ اﻟﺒﯿﺎﻧﻲ)‪. (C‬‬
      ‫6( اﻟﻨﻘﻄﺘﺎن اﻟﻤﺘﻨﺎﻇﺮﺗﺎن ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻨﻘﻄﺔ ‪ S‬ھﻤﺎ:‬              ‫} { = ) ‪(Cf ) ∩ ( d‬‬
                          ‫)0 ، 4( و )6- ، 2-(‬
                                                                    ‫2( ‪(Cf ) ∩ ( xx / ) =  − 1 ,0 ‬‬
                                                                                       ‫‪‬‬        ‫‪‬‬
              ‫‪x‬‬      ‫∞-‬                    ‫∞+‬      ‫15 1(‬                               ‫‪ 2  ‬‬
           ‫)‪f'(x‬‬                                                    ‫})1,0({ = ) / ‪(Cf ) ∩ ( yy‬‬
                                            ‫∞+‬                                                                          ‫94‬
                                                             ‫1( اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﻘﺎرب اﻟﻤﺎﺋﻞ ﻣﻌﺎدﻟﺘﮫ 2-‪y=x‬‬
           ‫)‪f(x‬‬
                                                             ‫اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﻘﺎرب اﻟﻌﻤﻮدي ﻣﻌﺎدﻟﺘﮫ 2-=‪x‬‬
                     ‫∞-‬                                            ‫2( })9-,3-({=)‪.(Cf)∩(Cg‬‬
                                                      ‫2(‬                                                          ‫05 1(‬
          ‫∞- ‪x‬‬                    ‫0‬          ‫∞+‬
                                                                                    ‫881‬                 ‫594‬
        ‫)‪g'(x‬‬                                                 ‫∞- ‪x‬‬              ‫−‬
                                                                                    ‫792‬                 ‫881‬
                                                                                                                   ‫∞+‬
                            ‫∞+‬                ‫∞+‬             ‫)‪f'(x‬‬                   ‫0‬                  ‫0‬
         ‫)‪g(x‬‬                                                                       ‫194‬                           ‫∞+‬
                                                                                    ‫68‬
                                      ‫∞-‬                     ‫)‪f(x‬‬
                ‫∞-‬
                                                                                                       ‫7001‬
                             ‫‪y‬‬                                                         ‫−‬
                             ‫3‬                                  ‫∞-‬                        ‫68‬

                                                             ‫2( ﺳﺒﻖ اﻟﺘﻄﺮق إﻟﻰ ﻛﯿﻔﯿﺔ إﺛﺒﺎت ﻣﺮﻛﺰ اﻟﺘﻨﺎﻇﺮ.‬
                             ‫2‬                                               ‫‪y‬‬                        ‫3(‬
                                                                                      ‫5‬
                                                                                      ‫4‬
                             ‫1‬                                                        ‫3‬
                                                                                      ‫2‬
                                                                                      ‫1‬

                     ‫1-‬      ‫0‬             ‫‪1 x‬‬                         ‫0 1- 2- 3-‬             ‫1‬    ‫2‬    ‫3‬     ‫4‬   ‫‪5 x‬‬
                                                                              ‫1-‬
                                                                              ‫2-‬
                            ‫1-‬                                                ‫3-‬
                                                                              ‫4-‬
                                                                              ‫5-‬
                            ‫2-‬                                                ‫6-‬
                                                                              ‫7-‬
                                                                              ‫8-‬
                            ‫3-‬                                                ‫9-‬
                                                                             ‫01-‬
                                                                             ‫11-‬
                            ‫4-‬
                                                                                                                     ‫4(‬
                                           ‫3( )‪.y=5x :(d‬‬                                                           ‫∞+‬
                                                                  ‫‪x‬‬        ‫∞-‬
         ‫‪x‬‬         ‫∞−‬         ‫0‬             ‫∞+‬                  ‫)‪f'(x‬‬
          ‫1‬                                                                ‫1-‬                      ‫∞+‬
        ‫−‬               ‫+‬                    ‫−‬
          ‫‪x‬‬                                                         ‫)‪f(x‬‬
      ‫اﻟﻮﺿﻌﯿﺔ‬          ‫)‪(Cg‬‬                ‫()‪Cg‬‬
                                                                                              ‫∞-‬                     ‫1-‬
                   ‫ﻓﻮق اﻟﻤﺴﺘﻘﯿﻢ‬        ‫ﺗﺤﺖ اﻟﻤﺴﺘﻘﯿﻢ‬
                                                                                          ‫‪y‬‬                         ‫5(‬
                                                                                          ‫4‬
                ‫4( })4,1(, )4-,1-({=)‪(Cf)∩(Cg‬‬                                             ‫3‬
                                                                                          ‫2‬
                                                                                          ‫1‬
     ‫25 1( ﺳﺒﻖ ﻛﯿﻔﯿﺔ إﺛﺒﺎت وﺟﻮد ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﻣﺎﺋﻞ‬                          ‫0 1- 2- 3- 4-‬          ‫‪1 2 3 4 x‬‬
                    ‫و دراﺳﺔ اﻟﻮﺿﻌﯿﺔ اﻟﻨﺴﺒﯿﺔ.‬                                         ‫1-‬
                                                                                     ‫2-‬
                                                                                     ‫3-‬
                                                                                     ‫4-‬




‫11‬
                                           ‫‪http://assil.yoo7.com‬‬
                                                        ‫6(‬                                                          ‫2(‬
                              ‫‪y‬‬
                              ‫6‬                                ‫∞- ‪x‬‬           ‫1-‬            ‫0‬            ‫1‬        ‫∞+‬
                                                             ‫)‪f'(x‬‬              ‫0‬                         ‫0‬
                              ‫5‬                                                        ‫∞+‬
                                                                  ‫∞+‬                                     ‫1-‬
                              ‫4‬                              ‫)‪f(x‬‬
                                                                              ‫3‬                 ‫∞-‬                 ‫∞-‬
                              ‫3‬
                                                                                        ‫‪y‬‬
                                                                                        ‫5‬
                              ‫2‬
                                                                                        ‫4‬
                                                                                        ‫3‬
                                                                                        ‫2‬
                               ‫2‬               ‫2‬        ‫45‬
                     ‫‪‬‬    ‫‪3  1 3‬‬                                                     ‫1‬
           ‫1( ‪f ( x ) =  x −  +  + ‬‬
                     ‫‪‬‬    ‫‪2  x 2‬‬                                     ‫0 1- 2- 3- 4-‬            ‫‪1 2 3 4 x‬‬
                                                                                  ‫1-‬
                                                    ‫2(‬
                                                                                  ‫2-‬
                     ‫1‬
       ‫∞- ‪x‬‬      ‫−‬
                     ‫2‬         ‫0‬       ‫2‬           ‫∞+‬                             ‫3-‬
     ‫)‪f'(x‬‬        ‫0‬                    ‫0‬
           ‫∞+‬                ‫∞+ ∞+‬                 ‫∞+‬
                                                                              ‫3( ﻟﻤﺎ [1,1-]∈‪ m‬ﻻ ﯾﻮﺟﺪ ﺣﻠﻮل.‬
     ‫)‪f(x‬‬                                                                     ‫ﻟﻤﺎ 1-=‪ m‬ﺣﻞ ﻣﻀﺎﻋﻒ 1=‪.x‬‬
                                         ‫71‬
                  ‫71‬
                   ‫2‬                      ‫2‬                                    ‫ﻟﻤﺎ 1=‪ m‬ﺣﻞ ﻣﻀﺎﻋﻒ 1-=‪.x‬‬
                                                                      ‫ﻟﻤﺎ [∞+,1]∪[1-,∞-]∈‪ m‬ﯾﻮﺟﺪ ﺣﻠﯿﻦ.‬
                                                        ‫3(‬
                  ‫‪y‬‬                                                                          ‫‪ − m +1 ‬‬
                  ‫8‬                                                                         ‫‪I‬‬           ‫4( ‪, m ‬‬
                                                                                             ‫2 ‪‬‬             ‫‪‬‬
                  ‫7‬                                                      ‫5( اﻟﻤﻤﺎس ﯾﻮازي ﻣﺤﻮر اﻟﻔﻮاﺻﻞ ﻣﻌﻨﺎه:‬
                                                                                            ‫0=)0‪ f /(x‬و ﻣﻨﮫ:‬
                  ‫6‬                                                                 ‫)1- , 1(‪.A(-1 , 3) , B‬‬
                                                                                ‫‪ B، A‬و ‪ I‬ﻓﻲ اﺳﺘﻘﺎﻣﯿﺔ ﻣﻌﻨﺎه:‬
                                                                                                              ‫→‬    ‫→‬
                  ‫5‬
                                                                           ‫‪ AB , AI‬ﻣﺘﻮازﯾﺎن. و ھﺬا ﻣﺤﻘﻖ.‬
                  ‫4‬
                                                        ‫4(‬                    ‫35 1( ﻟﯿﻜﻦ ‪ x∈D‬و ‪:-x∈D‬‬
                     ‫−‬
                         ‫1‬
                                   ‫0‬                                ‫ﻟﺪﯾﻨﺎ )‪ f ( − x) = f ( x‬إذن ‪ f‬زوﺟﯿﺔ.‬
       ‫∞- ‪x‬‬              ‫2‬                 ‫2‬        ‫∞+‬
      ‫‪g'(x‬‬                                                                                                        ‫2(‬
                         ‫0‬                 ‫0‬
           ‫∞+‬                                       ‫∞+‬                 ‫‪x‬‬        ‫0‬                    ‫2‬             ‫∞+‬
      ‫)‪g(x‬‬                                                            ‫)‪f'(x‬‬                        ‫0‬
                     ‫431‬                 ‫431‬                                                                      ‫∞+‬
                                                                                  ‫∞+‬
                      ‫56‬                  ‫56‬
                                                                       ‫)‪f(x‬‬
         ‫5( اﻟﻤﺴﺎﻓﺔ ‪ AM‬ﻣﻤﺜﻠﺔ ﺑﺎﻟﺪاﻟﺔ ‪ g‬و ﺗﻜﻮن ﻟﮭﺎ ﻗﯿﻤﺔ‬                                             ‫9‬
       ‫6( ﯾﺘﻌﺎﻣﺪ اﻟﻤﻤﺎس ﻟـ )‪ (H‬ﻓﻲ اﻟﻨﻘﻄﺔ 1‪ M‬و اﻟﻤﺴﺘﻘﯿﻢ‬                                             ‫2‬
        ‫)1‪ (AM‬إدا ﻛﺎن ﺟﺪاء ﻣﻌﺎﻣﻠﻲ ﺗﻮﺟﯿﮭﮭﻤﺎ ﯾﺴﺎوي‬                     ‫3( ﻣﻌﺎدﻟﺔ اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﻘﺎرب ھﻲ: 0=‪.x‬‬
                    ‫1‬                                                                             ‫1‬
                  ‫1- و ھﺪا ﻣﺤﻘﻖ ﻷن: 1− = 4 × −‬                                           ‫4( 2 = ‪MN‬‬
                    ‫4‬                                                                            ‫‪x‬‬
                        ‫ﻧﻔﺲ اﻟﺸﺊ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﺤﺎﻟﺔ اﻟﺜﺎﻧﯿﺔ.‬              ‫0 = ‪. lim MN = 0 , lim MN‬‬
                                                                       ‫∞− → ‪x‬‬                   ‫∞+ → ‪x‬‬

                                                                                       ‫5( )‪ (C‬ﯾﻘﻊ أﻋﻠﻰ )‪. (P‬‬

‫21‬
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                   ‫ب/ دراﺳﺔ اﻟﻮﺿﻌﯿﺔ ﺗﺘﻢ ﻛﻤﺎ ﺳﺒﻖ.‬                                                                   ‫55‬

                                 ‫) ‪u( x‬‬            ‫36‬       ‫∞- ‪x‬‬              ‫2-‬        ‫0‬               ‫2‬        ‫∞+‬
                   ‫= 1− ) ‪f ( x‬‬       ‫2‬
                                         ‫1( ﻟﺪﯾﻨﺎ:‬
                                    ‫‪x‬‬                     ‫)‪f'(x‬‬                         ‫0‬
                               ‫1 ) ‪u( x‬‬                                   ‫∞+‬            ‫−‬
                                                                                            ‫1‬               ‫∞+‬
                        ‫و ﻛﺪﻟﻚ: ≤ 2 ≤ 0‬
                                ‫‪x‬‬        ‫‪x‬‬                ‫)‪f(x‬‬                              ‫4‬
                                           ‫1‬
                            ‫و ﻣﻨﮫ: ≤ 1 − ) ‪f ( x‬‬                               ‫-∞‬                  ‫-∞‬              ‫1‬
                                           ‫‪x‬‬                     ‫1‬
                                   ‫1‬
      ‫2( ﺑﻤﺎأن: 0 = ‪ lim‬ﻓﺈن: 1 = ) ‪lim f ( x‬‬                                            ‫65 1( ‪Df=ℜ‬‬
     ‫∞+ → ‪x‬‬                ‫‪x → +∞ x‬‬
                                                           ‫2( اﻧﻄﻼﻗﺎ ﻣﻦ 1+ ≤ ‪ -1≤ cos x‬ﯾﻤﻜﻦ ﺣﺼﺮ‬
                                                                 ‫)‪ f(x‬ﺛﻢ اﻹﺣﺎﺑﺔ ﻋﻠﻰ اﻟﺴﺆال 3(.‬
                                                  ‫46‬
 ‫1( ∞+ = ) ‪lim f ( x ) = −∞ , lim f ( x‬‬                                            ‫75 1( 1=‪.y=3 , y=-2 , x‬‬
 ‫∞− → ‪x‬‬                   ‫∞+ → ‪x‬‬
                                                                                              ‫2( ﯾﺘﻢ اﻟﺮﺳﻢ.‬
                ‫2( ﻣﻦ أﺟﻞ ﻧﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪:x‬‬
                    ‫2 − ‪f / (x) = x2 − x‬‬
       ‫ﻟﻤﺎ [∞+,2]∪[1-,∞-]∈‪ x‬ﻓﺈن:0>)‪f /(x‬‬                                              ‫85 1( }1{- ‪= ℜ‬‬
                                                                                         ‫‪.Df‬‬
                 ‫ﻟﻤﺎ [2,1-]∈‪ x‬ﻓﺈن: 0<)‪f /(x‬‬                  ‫∞− = ) ‪lim f ( x ) = +∞ , lim f ( x‬‬
                                         ‫3(‬                 ‫∞+ → ‪x‬‬              ‫∞− → ‪x‬‬
                                                                                                 ‫2 ( أ/‬
                                                            ‫∞− = ) ‪lim f ( x‬‬
      ‫∞- ‪x‬‬           ‫1-‬           ‫2‬       ‫∞+‬                ‫1→ ‪x‬‬

     ‫)‪f'(x‬‬            ‫0‬                ‫0‬                                ‫ب/ 1=‪ x‬ﻣﻌﺎدﻟﺔ اﻟﻤﻘﺎرب اﻟﻌﻤﻮدي.‬
                     ‫71‬                      ‫∞+‬                                       ‫3( ﺗﺼﺤﯿﺢ: 1≠‪x‬‬
                     ‫21‬                                                     ‫2-=‪a=-1 , b=0 , c‬‬
     ‫)‪f(x‬‬                              ‫72‬                             ‫4( ﻣﻌﺎدﻟﺔ اﻟﻤﻘﺎرب اﻟﻤﺎﺋﻞ ھﻲ:1-‪y=x‬‬
                                   ‫−‬
                                       ‫21‬                              ‫5( })3 , 2(,)1 , 0({=)‪(C)∩(d‬‬
           ‫∞-‬
             ‫4( ﺗﻢ اﻟﺘﻄﺮق ﻹﺛﺒﺎت ﻣﺮﻛﺰ اﻟﺘﻨﺎﻇﺮ.‬
                                                                               ‫1 + ‪ϕ(h)=h2 + 3h‬‬                  ‫95 1(‬
            ‫5( ﻟﻠﻤﻌﺎدﻟﺔ 0=)‪ f(x‬ﺛﻼث ﺣﻠﻮل ھﻲ:‬
                                                                       ‫1 = ) ‪lim f ( x ) = lim h( x‬‬              ‫2(‬
              ‫1‬         ‫3 3 +1‬        ‫3 3 +1‬                           ‫1→ ‪x‬‬            ‫0 →‪h‬‬
          ‫=‪x= ,x‬‬                 ‫=‪,x‬‬
              ‫2‬            ‫2‬               ‫2‬                                                            ‫06‬
                                              ‫6(‬                                 ‫ﺗﺼﺤﯿﺢ: اﻟﻤﻘﺎم ھﻮ: 2+‪x‬‬
                           ‫‪y‬‬                                                                         ‫1(‬
                             ‫2‬
                             ‫1‬                                       ‫∞+ = ) ‪lim f ( x ) = −∞ , lim f ( x‬‬
                                                             ‫2− →‪x ‬‬
                                                                ‫<‬
                                                                                                ‫2− →‪x ‬‬
                                                                                                   ‫>‬

                    ‫0 1- 2-‬        ‫‪1 2 3 x‬‬                                  ‫2( 3=‪a=2 , b=-1 , c‬‬
                        ‫1-‬
                                                                                         ‫3( 1 – ‪y=2x‬‬
                        ‫2-‬
                                                                                  ‫4( ﯾﻤﻜﻦ اﻟﺘﺤﻘﻖ ﻣﻦ دﻟﻚ.‬
                        ‫3-‬
                                                                           ‫5( ﯾﺘﻢ دراﺳﺔ اﻟﻮﺿﻌﯿﺔ ﻛﻤﺎ ﺳﺒﻖ.‬
                 ‫7( ﺳﺒﻖ اﻟﺘﻌﺮض ﻟﻤﺜﻞ ھﺪا اﻟﺴﺆال.‬                        ‫16 1( 1-=‪a=1 , b=0 , c=2 , d‬‬
                                                                                 ‫2( ‪x=1 , x=-1 , y=x‬‬
       ‫56 1( ﻧﻼﺣﻆ أﻧﮫ ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪: x‬‬                         ‫3( دراﺳﺔ اﻟﻮﺿﻌﯿﺔ ﺗﻢ اﻟﺘﻄﺮق ﻟﮭﺎ ﺳﺎﺑﻘﺎ.‬
       ‫‪ -x∈ℜ‬و )‪ f(-x)=-f(x‬إدن ‪ f‬ﻓﺮدﯾﺔ.‬
         ‫‪f ( x + 2π ) = x + 2π − sin x‬‬
                                                                 ‫1( 2 = ) ‪lim f ( x ) = 2 , lim f ( x‬‬                  ‫26‬
         ‫‪f ( x + 4π ) = x + 4π − sin x‬‬     ‫2(‬                    ‫∞− → ‪x‬‬                  ‫∞+ → ‪x‬‬

                                                                      ‫)‪ (C‬ﯾﻘﺒﻞ ﻣﺴﺘﻘﯿﻢ ﻣﻘﺎرب ﻣﻌﺎدﻟﺘﮫ:2=‪y‬‬
         ‫‪f ( x + k 2π ) = x + k 2π − sin x‬‬                               ‫2( أ/ 1-=‪a=2 , b=-3 , c‬‬




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                                                                                                            ‫3(‬
                                                                     ‫‪x‬‬      ‫0‬                          ‫‪2π‬‬
                           ‫}1-,1{-‪Df=ℜ‬‬      ‫76 1(‬                  ‫)‪f'(x‬‬
                           ‫‪x − 3x − 18 x‬‬
                             ‫4‬       ‫2‬
                                                                                                       ‫‪2π‬‬
                  ‫= )‪f /(x‬‬                  ‫5(‬
                               ‫2) 1 − 2 ‪( x‬‬                         ‫)‪f(x‬‬
                       ‫)6+‪P(x)=x(x-3)(x2+3x‬‬                                 ‫0‬

   ‫∞- ‪x‬‬         ‫1-‬          ‫0‬            ‫1‬    ‫3‬           ‫∞+‬             ‫ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪ x‬ﻟﺪﯾﻨﺎ:‬
‫)‪f'(x‬‬                      ‫0‬                  ‫0‬                ‫2 ≤ ) ‪ 0 ≤ f / ( x‬و ﻣﻨﮫ ‪ f‬ﻣﺘﺰاﯾﺪة ﻋﻠﻰ ‪.ℜ‬‬
             ‫∞+‬           ‫9-‬             ‫∞+‬               ‫∞+‬          ‫1 ≤ ‪− 1 ≤ − sin x‬‬
‫)‪f(x‬‬                                                                  ‫4( ‪x − 1 ≤ x − sin x ≤ 1 + x‬‬
                                                  ‫9‬
       ‫∞-‬         ‫∞-‬                ‫∞-‬                                ‫‪x −1 ≤ f ( x ) ≤ 1+ x‬‬
                                                  ‫2‬
                                                                                                ‫5( ﻟﺪﯾﻨﺎ:‬
                                                                     ‫‪ x −1 ≥ A‬و 1 − ‪f ( x ) ≥ x‬‬
                                     ‫86 ‪Df= ℜ (1 (Ι‬‬
                                                                                       ‫إدن: ‪f ( x ) ≥ A‬‬
                       ‫‪x‬‬
        ‫= ) ‪f( x‬‬          ‫ﻟﻤﺎ [∞+,0[∈‪ x‬ﻓﺈن:‬                      ‫ﺣﺴﺐ ﺗﻌﺮﯾﻒ اﻟﻨﮭﺎﯾﺔ ﻟﻤﺎ ‪ x‬ﯾﺆول إﻟﻰ ∞+‬
                     ‫1+ ‪x‬‬
                       ‫‪x‬‬                                                         ‫ﻓﺈن: ∞+ = ) ‪lim f ( x‬‬
                                                                                    ‫∞+ → ‪x‬‬
            ‫= ) ‪f( x‬‬       ‫ﻟﻤﺎ ]0,∞-]∈‪ x‬ﻓﺈن:‬
                     ‫‪1− x‬‬                                                             ‫‪y‬‬
                                                                                      ‫5‬
                                 ‫2( ‪ f‬داﻟﺔ ﻓﺮدﯾﺔ.‬                                     ‫4‬
                              ‫3( 1 = ) ‪lim f ( x‬‬                                      ‫3‬
                                ‫∞+ → ‪x‬‬                                                ‫2‬
                       ‫4( ﻟﻤﺎ [∞+,0[∈‪ x‬ﻓﺈن:‬                                           ‫1‬
                                       ‫1‬
                         ‫= )‪f /(x‬‬                                     ‫0 1- 2- 3- 4- 5-‬
                                                                                   ‫1-‬
                                                                                             ‫5 4 3 2 1‬      ‫‪x‬‬
                                   ‫2) 1 + ‪( x‬‬                                      ‫2-‬
                          ‫)‪f(x‬‬             ‫) ‪f( x‬‬                                  ‫3-‬
                    ‫‪lim‬‬         ‫‪= lim‬‬             ‫1=‬                               ‫4-‬
                      ‫→‪‬‬
                  ‫0 <‪x ‬‬    ‫‪x‬‬         ‫→‪‬‬
                                   ‫0 >‪x ‬‬     ‫‪x‬‬                                    ‫5-‬
                        ‫و ﻣﻨﮫ ‪ f‬ﻗﺎﺑﻠﺔ ﻟﻺﺷﺘﻘﺎق ﻋﻨﺪ 0.‬                               ‫6-‬
                                                                                   ‫7-‬
                  ‫‪x‬‬      ‫∞- ‪‬‬                 ‫∞+ ‪‬‬
                                                                             ‫66 ﺗﺼﺤﯿﺢ: اﻟﻤﻘﺎم ھﻮ: ‪x-c‬‬
               ‫)‪f'(x‬‬                                               ‫1( ﻣﻌﺎدﻟﺔ اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﻘﺎرب ھﻲ:‪.x=c‬‬
                                                      ‫1‬                              ‫و ﻋﻠﯿﮫ 1=‪.c‬‬
                  ‫)‪f(x‬‬                                                                     ‫5‬
                                                                  ‫2( ﻟﺪﯾﻨﺎ: = ) 3 ( ‪ f‬و ﻣﻨﮫ:5=‪.6a+b‬‬
                          ‫1-‬                                                               ‫2‬
                                                                     ‫3( ﻟﺪﯾﻨﺎ: 0=)3(/ ‪ f‬و ﻣﻨﮫ: 0=‪.4a-b‬‬
                                                                                      ‫1‬         ‫2‬
                                                                            ‫+ ‪f( x ) = x‬‬            ‫4(‬
                                ‫‪y‬‬                         ‫5(‬                          ‫2‬      ‫1− ‪x‬‬
                                ‫1‬                                               ‫5( )‪ (Cf‬ﯾﻘﻊ أﻋﻠﻰ )‪(D‬‬

                ‫2-‬     ‫1-‬       ‫0‬        ‫1‬    ‫2‬       ‫‪x‬‬
                            ‫1-‬

                            ‫2-‬




                                                                       ‫1‬   ‫2‬    ‫3‬    ‫4‬    ‫5‬    ‫6‬   ‫7‬   ‫8‬    ‫‪x‬‬



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                                                 ‫‪y‬‬
                                            ‫= ‪.x‬‬      ‫6( ﻟﻤﺎ 0≥‪: y‬‬
                                                ‫‪1− y‬‬
                                                 ‫‪y‬‬
                                         ‫= ‪.x‬‬          ‫ﻟﻤﺎ 0≤‪: y‬‬
                                                ‫‪1+ y‬‬
                              ‫7( اﻟﺤﻞ اﻟﻮﺣﯿﺪ ﻋﻠﻰ ‪ ℜ‬ﻟﻠﻤﻌﺪﻟﺔ ‪ f(x)=y‬ھﻮ‬
                                                             ‫‪y‬‬
                                                     ‫= ‪.x‬‬
                                                           ‫‪1− y‬‬
                                                 ‫‪Dg=ℜ-{-1,1} (1 (ΙΙ‬‬
                          ‫‪x‬‬
                ‫= ) ‪g( x‬‬      ‫2( ﻟﻤﺎ ]0, 1-]∪[1-, ∞-]∈‪ x‬ﻓﺈن:‬
                        ‫‪1+ x‬‬
                             ‫‪x‬‬
                   ‫= ) ‪g( x‬‬       ‫ﻟﻤﺎ [∞+, 1]∪[1, 0[∈‪ x‬ﻓﺈن:‬
                            ‫‪1− x‬‬
                                          ‫3( ∞+ = ) ‪lim g ( x‬‬
                                             ‫1 →‪x ‬‬
                                                ‫<‬

                                                         ‫1‬
                                             ‫= )‪g/(x‬‬           ‫4(‬
                                                     ‫2) ‪(1 − x‬‬
                                      ‫) ‪g( x‬‬         ‫) ‪g( x‬‬
                                 ‫‪lim‬‬         ‫‪= lim‬‬          ‫1=‬
                                   ‫→‪‬‬
                               ‫0 <‪x ‬‬   ‫‪x‬‬        ‫→‪‬‬
                                              ‫0 >‪x ‬‬   ‫‪x‬‬
                                       ‫‪x‬‬     ‫1-‬               ‫1‬
                                    ‫)‪f'(x‬‬
                                                           ‫∞+‬
                                    ‫)‪f(x‬‬
                                            ‫∞-‬
                                             ‫‪y‬‬
                                                                    ‫5(‬

                                            ‫2‬

                                            ‫1‬


                                  ‫1-‬         ‫0‬          ‫‪1 x‬‬

                                            ‫1-‬

                                            ‫2-‬

                                            ‫3-‬

                           ‫ﻣﻦ أﺟﻞ ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪ x‬ﻣﻦ اﻟﻤﺠﺎل [1 ، 1-]:‬
                                                      ‫‪(f o g )( x ) = x‬‬
                                ‫6( ﻧﺴﺘﻨﺘﺞ أن اﻟﻤﻨﺤﻨﯿﯿﻦ ﻣﺘﻨﺎﻇﺮﯾﻦ ﺑﺎﻟﻨﺴﺒﺔ‬
                                                          ‫إﻟﻰ )‪. (D‬‬




‫51‬
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posted:4/7/2011
language:Arabic
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