Originally Posted by

**auslmar** The problem is as follows:

$\displaystyle \lim_{x\to\infty}(1+\frac{2}{x})^\frac{x}{5}$

What I've done:

$\displaystyle \lim_{x\to\infty}(1+\frac{2}{x})^\frac{x}{5} = $ $\displaystyle e^{\lim_{x\to\infty}\frac{x}{5}ln(1+\frac{2}{x})}$

So

$\displaystyle \lim_{x\to\infty}\frac{x}{5}ln(1+\frac{2}{x}) = \lim_{x\to\infty}\frac{ln(1+\frac{2}{x})}{\frac{5} {x}}$

Using L'Hospital's Rule

$\displaystyle = \lim_{x\to\infty}\frac{\frac{-2}{x^2+2x}}{\frac{-5}{x^2}}$

This the point where I get confused. It seems like this limit is indeterminable and it seems like if you keep doing L'Hospital's Rule that it only gets more complex only requires more and more applications of L'Hospital's. What am I doing wrong? Do I need to keep going? If anyone can point out my errs and shed some light on the correct path, I'd greatly appreciation.

Thanks for your consideration,

Austin Martin