# limite

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```					‫)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

```
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 views: 53 posted: 10/14/2012 language: pages: 2