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kolutastmr 2022-07-06 Answered
Evaluate
lim x x 2 ( 4 1 x 4 1 1 + x )
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Answers (2)

billyfcash5n
Answered 2022-07-07 Author has 17 answers
Write
4 t 4 t / ( t + 1 ) t 2 = 4 t / ( t + 1 ) 4 t 2 / ( t + 1 ) 1 t 2 t + 1 1 t + 1
and use the standard limit lim u 0 a u 1 u = log a to find
lim t 0 4 t 4 t / ( t + 1 ) t 2 = lim t 0 4 t / ( t + 1 ) lim t 0 4 t 2 / ( t + 1 ) 1 t 2 t + 1 lim t 0 1 t + 1 = 1 log 4 1 = log 4

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2nalfq8
Answered 2022-07-08 Author has 3 answers
Mean Value Theorem, for f ( x ) = 4 x , with f ( x ) = 4 x ln 4: There exists a ξ x ( 1 / x , 1 / ( x + 1 ) ) , such that
4 1 x 4 1 x + 1 = ( 1 x 1 x + 1 ) 4 ξ x ln 4 = 4 ξ x ln 4 x ( x + 1 )
Clearly lim x ξ x = 0, and hence lim x 4 ξ x = 1. Hence
lim x x 2 ( 4 1 x 4 1 x + 1 ) = lim x x 2 x ( x + 1 ) 4 ξ x ln 4 = ln 4.

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