1. Exponential Limit

Hi all,

I'm trying to evaluate the limit of (a(1-exp(a)))/(n(1-exp(a/n))) as n goes to infinity and 'a' is constant.

I think that you have to rearrange and then use l'hoptials rule somehow, but i'm unsure. Any help would be greatly appreciated.

Thanks.

2. Write it as $\frac{f(x)}{g(x)}$ where $f(x)={\frac{a(1-e^a)}{n}}$ and $g(x)=e^{1- \displaystyle\frac{a}{n}}$.

3. Originally Posted by DeFacto
Hi all,

I'm trying to evaluate the limit of (a(1-exp(a)))/(n(1-exp(a/n))) as n goes to infinity and 'a' is constant.

I think that you have to rearrange and then use l'hoptials rule somehow, but i'm unsure. Any help would be greatly appreciated.

Thanks.
$\frac{a(1-e^a)}{n(1-e^{a/n})} = \frac{\frac{a(1-e^a)}{n}}{1-e^{a/n}} = \frac{f(n)}{g(n)}$

Use L'Hopital's Rule:

Spoiler:
$\frac{f'(n)}{g'(n)} = \frac{-\frac{a(1-e^a)}{n^2}}{-e^{a/n}\cdot -a/n^2} = -\frac{a(1-e^a)}{ae^{a/n}}$

$\lim_{n\to\infty}-\frac{a(1-e^a)}{ae^{a/n}} = -\frac{a(1-e^a)}{a} = \boxed{e^a-1}$