Docstoc

limite

Document Sample
limite Powered By Docstoc
					‫)ﻣﺤﻤﺪ اﻟﻜﯿﺎل(‬                              ‫اﻟﻨﮫﺎﻳﺎت‬

                            ‫‪Ë‬ﻧﮫﺎﻳﺎت اﻟﺪوال ‪ ( n Î¥ *) x a x n‬و ‪ x a x‬و ﻣﻘﻠﻮﺑﺎﺗﮫﺎ:‬

                      ‫0 = ‪lim x‬‬                                 ‫0 = ‪lim x n‬‬
                      ‫0®‪x‬‬                                       ‫0® ‪x‬‬
                       ‫>‬

                                ‫¥+ = ‪x‬‬                                     ‫1‬
                   ‫‪lim‬‬                                          ‫‪lim‬‬             ‫0=‬
                  ‫¥+® ‪x‬‬                                        ‫‪x ®-¥ x n‬‬
                                ‫1‬
                      ‫‪lim‬‬          ‫0=‬                           ‫‪lim‬‬
                                                                           ‫1‬
                                                                                ‫0=‬
                   ‫¥+® ‪x‬‬         ‫‪x‬‬                             ‫‪x ®+¥ x n‬‬


              ‫إذا ﻛﺎن ‪ n‬ﻋﺪدا ﻓﺮدﻳﺎ ﻓﺈن:‬                   ‫إذا ﻛﺎن ‪ n‬ﻋﺪدا زوﺟﯿﺎ ﻓﺈن:‬
                   ‫¥+ = ‪lim x n‬‬                                ‫¥+ = ‪lim x n‬‬
                  ‫¥+® ‪x‬‬                                       ‫¥+® ‪x‬‬
                   ‫¥- = ‪lim x‬‬   ‫‪n‬‬
                                                               ‫¥+ = ‪lim x n‬‬
                  ‫¥-® ‪x‬‬                                       ‫¥-® ‪x‬‬
                            ‫1‬                                          ‫1‬
                   ‫‪lim‬‬      ‫‪n‬‬
                                    ‫¥+ =‬                       ‫‪lim‬‬         ‫¥+ =‬
                   ‫‪x ®0 x‬‬                                      ‫‪x ® 0 xn‬‬
                     ‫>‬                                           ‫>‬
                            ‫1‬                                          ‫1‬
                   ‫‪lim‬‬      ‫‪n‬‬
                                    ‫¥- =‬                       ‫‪lim‬‬         ‫¥+ =‬
                   ‫‪x ®0 x‬‬                                      ‫‪x ® 0 xn‬‬
                     ‫<‬                                           ‫<‬


                ‫‪Ë‬ﻧﮫﺎﻳﺎت اﻟﺪوال اﻟﺤﺪودﻳﺔ و اﻟﺪوال اﻟﺠﺬرﻳﺔ ﻋﻨﺪ ¥+ أو ﻋﻨﺪ ¥- :‬
       ‫ﻧﮫﺎﻳﺔ داﻟﺔ ﺟﺬرﻳﺔ ﻋﻨﺪ ¥+ أو ﻋﻨﺪ ¥-‬              ‫ﻧﮫﺎﻳﺔ ﺣﺪودﻳﺔ ﻋﻨﺪ ¥+ أو ﻋﻨﺪ ¥- ھﻲ‬
           ‫ھﻲ ﻧﮫﺎﻳﺔ ﺧﺎرج ﺣﺪﻳﮫﺎ اﻷﻛﺒﺮ درﺟﺔ‬                             ‫ﻧﮫﺎﻳﺔ ﺣﺪھﺎ اﻷﻛﺒﺮ درﺟﺔ‬


                                                                 ‫‪Ë‬ﻧﮫﺎﻳﺎت اﻟﺪوال اﻟﻤﺜﻠﺜﯿﺔ:‬
             ‫1 ‪1 - cos x‬‬                       ‫‪tan x‬‬                           ‫‪sin x‬‬
        ‫‪lim‬‬           ‫=‬                    ‫‪lim‬‬       ‫1=‬                    ‫‪lim‬‬       ‫1=‬
        ‫0® ‪x‬‬     ‫²‪x‬‬     ‫2‬                  ‫‪x ®0 x‬‬                          ‫‪x ®0 x‬‬


                                                     ‫‪Ë‬ﻧﮫﺎﻳﺎت اﻟﺪوال ﻣﻦ اﻟﻨﻮع: ) ‪x a u ( x‬‬

                ‫‪lim‬‬      ‫)‪u(x‬‬                                           ‫) ‪lim u ( x‬‬
               ‫0‪x ® x‬‬                                                  ‫0‪x ® x‬‬


                        ‫‪l‬‬                                                      ‫0³ ‪l‬‬
                      ‫¥+‬                                                       ‫¥+‬
 ‫ھﺬه اﻟﻨﮫﺎﻳﺎت ﺗﺒﻘﻰ ﺻﺎﻟﺤﺔ ﻋﻨﺪ 0 ‪ x‬ﻋﻠﻰ اﻟﯿﻤﯿﻦ أو ﻋﻨﺪ 0 ‪ x‬ﻋﻠﻰ اﻟﯿﺴﺎر أو ﻋﻨﺪ ¥+ أو ﻋﻨﺪ ¥-‬

                                                 ‫6‬
                                                                                  :‫اﻟﻨﮫﺎﻳﺎت و اﻟﺘﺮﺗﯿﺐ‬Ë
                             ü
  u ( x ) £ f ( x ) £ V ( x )ï
                             ï                            f ( x ) - l £ V ( x )ü
                             ï                                                 ï
   lim u ( x ) = l           ý Þ lim f ( x ) = l                                 Þ lim f ( x ) = l
  x ® x0                                                   lim V ( x ) = 0 ý x ® x 0
                             ï x ® x0                     x ® x0               ï
                                                                               þ
   lim V ( x ) = l           ï
  x ® x0                     ï
                             þ
   u(x) £ V(x)       ü                                   u(x) £ f (x)      ü
                     ï                                                     ï
    lim V ( x ) = -¥ ý x ® x 0 ( )                        lim u ( x ) = +¥ ý x ® x 0 ( )
                       Þ lim f x = -¥                                        Þ lim f x = +¥
   x ® x0            ï
                     þ                                   x ® x0            ï
                                                                           þ
-¥ ‫ ﻋﻠﻰ اﻟﯿﺴﺎر أو ﻋﻨﺪ ¥+ أو ﻋﻨﺪ‬x 0 ‫ ﻋﻠﻰ اﻟﯿﻤﯿﻦ أو ﻋﻨﺪ‬x 0 ‫ھﺬه اﻟﻨﮫﺎﻳﺎت ﺗﺒﻘﻰ ﺻﺎﻟﺤﺔ ﻋﻨﺪ‬

                                                                            :‫اﻟﻌﻤﻠﯿﺎت ﻋﻠﻰ اﻟﻨﮫﺎﻳﺎت‬Ë
                                                                              :‫ﻧﮫﺎﻳﺔ ﻣﺠﻤﻮع داﻟﺘﯿﻦ‬

             lim f ( x )              l         l             l             -¥          +¥                +¥
            x ® x0
            lim g ( x )              l'        -¥             +¥            -¥          +¥                -¥
           x ® x0

    lim éf ( x ) + g ( x ) ù
        ë                  û    l + l'         -¥             +¥            -¥          +¥            ‫شغ م‬
   x ® x0

                                                                                  :‫ﻧﮫﺎﻳﺔ ﺟﺪاء داﻟﺘﯿﻦ‬

            lim f ( x )          l             l<0            l>0            -¥     -¥           +¥        0
          x ® x0
            lim g ( x )          l'       -¥        +¥    -¥       +¥        -¥     +¥           +¥       ±¥
          x ® x0

   lim éf ( x ) ´ g ( x ) ù
       ë                  û    l ´ l'     +¥        -¥    -¥       +¥        +¥     -¥           +¥
                                                                                                          ‫شغ‬
  x ® x0                                                                                                   ‫م‬

                                                                                  :‫ﻧﮫﺎﻳﺔ ﺧﺎرج داﻟﺘﯿﻦ‬

 lim f ( x )         l     l          l<0           l>0                -¥              +¥             0        ±¥
x ® x0

 lim g ( x ) l' ¹ 0 ±¥
x ® x0                           0-       0+    0-       0+       0-    0+        0-        0+        0        ±¥
         f (x)       l          +¥ -¥ -¥ +¥ +¥ -¥ -¥ +¥                                           ‫شغ‬           ‫شغ‬
 lim                       0
x ® x0   g(x)        l'                                                                            ‫م‬            ‫م‬

                                                                                             ‫ﻣﻼﺣﻈﺔ ﻋﺎﻣﺔ‬
-¥ ‫ ﻋﻠﻰ اﻟﯿﺴﺎر أو ﻋﻨﺪ ¥+ أو ﻋﻨﺪ‬x 0 ‫ ﻋﻠﻰ اﻟﯿﻤﯿﻦ أو ﻋﻨﺪ‬x 0 ‫ھﺬه اﻟﻨﮫﺎﻳﺎت ﺗﺒﻘﻰ ﺻﺎﻟﺤﺔ ﻋﻨﺪ‬



                                                     7

				
DOCUMENT INFO
Shared By:
Tags:
Stats:
views:53
posted:10/14/2012
language:
pages:2