Math Devoir 8

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ò‹Ñïå‚@óibïä

‘ì‹«@‹Ð

†bîŒ@æi@׊b€@óîíäbq

2HZŒb−fia@ò‡€€€€€€€€€€€€€€€€€€à

Zßìÿa@æî‹ánÜa

sin ( x )

 x2 −1  x +1 lim 3 ، lim Arc tan  Arc tan  2  :‫ﺃﺤﺴﺏ ﺍﻝﻨﻬﺎﻴﺎﺕ ﺍﻝﺘﺎﻝﻴﺔ‬  ، xlim x →0 →+∞   x − 1 + 1 x x + 1 − 1 x →1+     . lim

x →+∞

(

3

x 3 −1− x

)

ZðäbrÜa@æî‹ánÜa

 f ( x ) = 3 x + 1 + x ; x ∈ [ 0, +∞[    π  :‫ ﺍﻝﻤﻌﺭﻓﺔ ﺒﻤﺎﻴﻠﻲ‬f ‫ﻨﻌﺘﺒﺭ ﺍﻝﺩﺍﻝﺔ ﺍﻝﻌﺩﺩﻴﺔ‬ 2 f x x x = 1 + tan ; ∈ ( )  ( )  − 2 , 0   .‫ ﻓﻲ ﺍﻝﺼﻔﺭ‬f ‫ ﺃﺩﺭﺱ ﺍﺘﺼﺎل ﺍﻝﺩﺍﻝﺔ‬-1 . lim + f ( x ) ‫ ﻭ‬lim f ( x ) ‫ ﺃﺤﺴﺏ‬-2 x →−

.‫ﻴﺠﺏ ﺘﺤﺩﻴﺩﻩ‬

x →+∞

π

2

 π   π  J ‫ ﻨﺤﻭ ﻤﺠﺎل‬ − , 0  ‫ ﺘﻘﺎﺒل ﻤﻥ‬ − , 0  ‫ ﻋﻠﻰ ﺍﻝﻤﺠﺎل‬f ‫ ﻗﺼﻭﺭ ﺍﻝﺩﺍﻝﺔ‬g ‫ ﺒﻴﻥ ﺃﻥ‬-3  2   2  . J ‫ﻤﻥ‬

x ‫ ﻝﻜل‬g −1 ( x ) ‫ ﺤﺩﺩ‬-4 ZsÜbrÜa@æî‹ánÜa

u0 = 0   :   ‫( ا‬u n ) 

‫  ا‬  un2 = 8 + ; ∀ ∈ ℕ u n  n +1 3  . 0 ≤ u n ≺ 2 3; ∀n ∈ ℕ :‫   

 أن‬-1 .‫ " أ!  ر‬# $‫ ا‬%& '( )‫*ا‬+ (u n ) ‫  أن‬-2

(v n ) 

‫   ا‬-3 $)#‫( ه‬v n ) 

‫  أن ا‬:‫أ‬

.v n = 12 − u n :   ‫ا‬ 2

.‫ول‬.‫)ه ا‬0‫! و‬$ $‫)دا أ‬2 . n 6)

u n " # $‫ ا‬%& n 6) v n 450‫ أ‬:‫ب‬ . lim (u n ) 450‫ أ‬:‫ج‬

.S n = u 0 + u + ⋅ ⋅ ⋅ + u 2

2 1

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n →+∞ 2 n −1 :‫ع‬9:‫ا‬

n 6) 450‫ أ‬:‫د‬