# Lim(x->inf) of ((x+a)/(x-a))^x = e

#### techtown

lim(x-->inf) of ((x+a)/(x-a))^x = e

I started this problem and quickly became stuck, the question asks for what value of "a" is the following true:

lim(x-->inf) of ((x+a)/(x-a))^x = e

I took the natural log of both sides to start and got this:

lim(x-->inf) of x*ln((x+a)/(x-a)) = 1

I've tried going on from here but nothing in the end makes sense and i don't know any other way to start the problem; any help is appriciated, thanks.

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#### Maxos

The text is wrong:

$$\lim_{\substack{x\rightarrow 0}}f(x) = 1 , \forall a \in \mathbb{R}$$

whereas

$$\lim_{\substack{x\rightarrow \infty}}f(x) = e^{2a}$$

Ok?

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#### techtown

ah, yes, i did mean for x to go to infinity; but how did you get e^2a?

#### Maxos

$$\lim_{\substack{ x \rightarrow \infty}} {(\frac {x+a}{x-a})}^x = \lim_{\substack{ x \rightarrow \infty}} {(1+ \frac {2a}{x-a})}^x = \\ \lim_{\substack{ y \rightarrow \infty}} {(1+ \frac {2a}{y})}^{y+a}=$$
$$\lim_{\substack{y \rightarrow \infty}} {(1+ \frac {2a}{y})}^y {(1+ \frac {2a}{y})}^a = \\ \lim_{\substack{y\rightarrow \infty}}{(1+ \frac {2a}{y})}^y = e^{2a}$$

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#### techtown

thank you, i think i have it now

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