- #1

James Brady

- 105

- 4

## Homework Statement

Find ##\lim_{x\to\infty} x(e^{1/x}-1)##

## Homework Equations

##\lim_{x\to\infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \frac{f'(x)}{g'(x)}##

## The Attempt at a Solution

I attempted to rewrite the function in terms of a ratio and then use L'Hopital's rule:

##\lim_{x\to\infty} \frac{x}{(e^{1/x}-1)^{-1}} = \lim_{x\to\infty} \frac{1}{-(e^{1/x}-1)^{-2}(\frac{1}{x}e^{1/x})}##

The problem is that the exponential terms never go away. The bigger problem is I believe L'Hopital's Rule is probably unnecessary and I'm missing something more basic here.