Saw this problem and I thought I'd take a shot at it: lim_{x->oo}((x+a)/(x-a))^x = e.

Grayson Pierce 2022-07-17 Answered
Find all a such that lim x ( x + a x a ) x = e
Saw this problem and I thought I'd take a shot at it:
Find all a such that
lim x ( x + a x a ) x = e .
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Answers (2)

Coleman Ali
Answered 2022-07-18 Author has 13 answers
Note that
x + a x a = 1 + a x 1 a x .
So if you can show (or simply recognize from theorem) that
lim x ( 1 + a x ) x = e a , lim x ( 1 a x ) x = e a ,
then you get that a ( a ) = 1 so a = 1 / 2

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Nash Frank
Answered 2022-07-19 Author has 10 answers
Let
L = lim x ( x + a x a ) x .
Then
log L = lim x x ( log ( x + a ) log ( x a ) ) = LHR lim x 1 / ( x + a ) 1 / ( x a ) 1 / x 2 = lim x x 2 ( x + a ) x 2 ( x a ) ( x + a ) ( x a ) = 2 a . log L = 2 a L = e 2 a .
So it is true only for a = 1 / 2

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