Math Devoir 2

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‫ﻧﻴﺎﺑﺔ‪ ‬ﻛﻠﻤﻴﻢ‪ ‬ﺑﺎﺏ‪ ‬ﺍﻟﺼﺤﺮﺍء‬ ‫ﺛﺎﻧﻮﻳﺔ‪ ‬ﺍﻟﺨﻮﺍﺭﺯﻣﻲ‪                       ‬ﻓﺮﺽ‪ ‬ﻣﺤﺮﻭﺱ‪ ‬ﺭﻗﻢ‪2- 1‬ﺑﺎﻙ‪                                       -1 ‬ﺫ‪ :‬ﺍﻟﺰﻏﺪﺍﻧﻲ‬ ‫ﺍﻟﺘﻤﺮﻳﻦ‪ ‬ﺍﻷﻭﻝ‪:‬‬

‫‪‬‬

‫ﺃﺣﺴﺐ‪ ‬ﺍﻟﻨﻬﺎﻳﺎﺕ‪ ‬ﺍﻟﺘﺎﻟﻴﺔ‪: ‬‬

‫‪1  x3  4 x‬‬

‫‪3‬‬

‫‪‬‬

‫‪x  x 1‬‬ ‫‪x x 2  7‬‬ ‫‪2‬‬

‫‪lim‬‬

‫‪x ‬‬

‫‪,‬‬

‫‪lim‬‬

‫ﺍﻟﺘﻤﺮﻳﻦ‪ ‬ﺍﻟﺜﺎﻧﻲ‪:‬‬ ‫ﻧﻌﺘﺒﺮ‪ ‬ﺍﻟﻤﺘﺘﺎﻟﻴﺘﻴﻦ‪ (Un) ‬ﻭ)‪ (Vn‬ﺍﻟﻤﻌﺮﻓﺘﻴﻦ‪ ‬ﻛﻤﺎ‪ ‬ﻳﻠﻲ‬

‫‪ U n 1  U n 32 V n‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ V n 1  U n  3 V n‬‬ ‫‪4‬‬ ‫‪‬‬

‫‪ (1‬ﻧﻀﻊ ‪  n1 Vn Un‬ﺑﺮﻫﻦ‪ ‬ﺃﻥ) ‪(  n‬ﻫﻨﺪﺳﻴﺔ‪ ‬ﻭﺣﺪﺩ‪ ‬ﺣﺪﻫﺎ‪ ‬ﺍﻷﻭﻝ‪ ‬ﻭ‪ ‬ﺃﺳﺎﺳﻬﺎ‪: ‬‬ ‫‪ (2‬ﺍﻋﻂ‪ ‬ﺻﻴﻐﺔ ‪   n‬ﺑﺪﻻﻟﺔ‪  n‬ﺛﻢ‪ ‬ﺍﺳﺘﻨﺘﺞ‪ ‬ﺃﻧﻪ‪n   : Vn  U n  0  ‬‬ ‫‪ (3‬ﺑﺮﻫﻦ‪ ‬ﺃﻥ‪ ‬ﺍﻟﻤﺘﺘﺎﻟﻴﺔ)‪ (Un‬ﺗﺰﺍﻳﺪﻳﺔ‪ ‬ﻭﺃﻥ‪ (Vn) ‬ﺗﻨﺎﻗﺼﻴﺔ‪. ‬‬

‫‪ (4‬ﺑﺮﻫﻦ‪ ‬ﺃﻧﻪ‪n   : U0  Un  Vn  V0  ‬‬ ‫‪ (5‬ﺃﺳﺘﻨﺘﺞ‪ ‬ﺃﻥ‪ ‬ﺍﻟﻤﺘﺘﺎﻟﻴﺘﺎﻥ‪ (Un) ‬ﻭ)‪ (Vn‬ﻣﺘﻘﺎﺭﺑﺘﺎﻥ‪. ‬‬

‫‪ (6‬ﺃﺳﺘﻨﺘﺞ‪ ‬ﺣﺴﺐ‪ ‬ﻣﺎ‪ ‬ﺳﺒﻖ‪ ‬ﺃﻥ‪ l i m V n : ‬‬ ‫‪n‬‬

‫‪n‬‬

‫‪ - l i nm U‬ﻧﺮﻣﺰ‪ ‬ﻟﻬﺪﻩ‪ ‬ﺍﻟﻨﻬﺎﻳﺔ‪ ‬ﺏ‪L ‬‬

‫‪ (7‬ﻧﻌﺘﺒﺮ‪ ‬ﺍﻟﻤﺘﺘﺎﻟﻴﺔ)‪ (tn‬ﺍﻟﻤﻌﺮﻓﺔ‪ ‬ﻛﻤﺎ‪ ‬ﻳﻠﻲ ‪:‬‬

‫‪ n   : t n  3U n  8V n‬‬

‫‪                   ‬ﺃ( ﺑﺮﻫﻦ‪ ‬ﺃﻧﻪ‪    n   : t n 1  t n    ‬ﺛﻢ‪ ‬ﺃﺳﺘﻨﺘﺞ‪ ‬ﺃﻧﻪ‪: ‬‬ ‫‪                 ‬ﺏ( ﺃﺳﺘﻨﺘﺞ‪ ‬ﻗﻴﻤﺔ ‪L :‬‬

‫‪n : tn  99‬‬

‫ﺍﻟﺘﻤﺮﻳﻦ‪ ‬ﺍﻟﺜﺎﻟﺚ‪:‬‬

‫( ‪f : x  A rcT a n‬‬

‫‪      ‬ﻧﻌﺘﺒﺮ‪ ‬ﺍﻟﺪﺍﻟﺔ‪ ‬ﺍﻟﻌﺪﺩﻳﺔ‪2 x  3 )  ‬‬

‫‪ (1‬ﺑﺮﻫﻦ‪ ‬ﺃﻥ‪ ‬ﻣﺠﻤﻮﻋﺔ‪ ‬ﺗﻌﺮﻳﻒ‪ ‬ﺍﻟﺪﺍﻟﺔ‪ f  ‬ﻫﻲ‬ ‫‪  f‬ﺗﺰﺍﻳﺪﻳﺔ‪ ‬ﻗﻄﻌﺎ‪ ‬ﻋﻠﻰ ‪D‬‬

‫‪3‬‬ ‫‪‬‬ ‫‪D   ;   :‬‬ ‫‪2‬‬ ‫‪‬‬

‫‪ ‬ﺛﻢ‪ ‬ﺃﺣﺴﺐ‪lim f ( x )  ‬‬

‫‪ (2‬ﺑﻴﻦ‪ ‬ﺃﻥ‬ ‫‪ (3‬ﺑﻴﻦ‪ ‬ﺃﻥ‪   f  ‬ﻣﺘﺼﻠﺔ‪ ‬ﻋﻠﻰ‪. D  ‬‬ ‫‪(4‬‬ ‫‪                  ‬ﺃ( ﺑﻴﻦ‪ ‬ﺃﻥ‪  f    ‬ﺗﻘﺎﺑﻞ‪ ‬ﻣﻦ‪    D   ‬ﻧﺤﻮ‪ ‬ﻣﺠﺎﻝ ‪    J‬ﻳﺠﺐ‪ ‬ﺗﺤﺪﻳﺪﻩ‪. ‬‬ ‫‪.‬‬

‫‪                ‬ﺏ( ﺃﺣﺴﺐ‪   f 1 ( y )  ‬ﻟﻜﻞ ‪  y‬ﻣﻦ‪. J  ‬‬

‫‪x  ‬‬

‫‪U01‬‬ ‫‪‬‬ ‫‪V012‬‬

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