# Math Help - Is the correct Limit?

1. ## Is the correct Limit?

lim as n approaches infinity of (n/(n+1))^2n

limit equals e?

if so why?

2. Originally Posted by Nichelle14
lim as n approaches infinity of (n/(n+1))^2n

limit equals e?

if so why?
No,

Consider,
$\lim_{n\to\infty}\left( 1+\frac{1}{n} \right)^n=e$
Since, the limit exists, then you can evaluate, $f(x)=1/x$ because it is countinous for $x>0$.
Thus,
$f\left( \lim_{n\to\infty}\left( 1+\frac{1}{n} \right)^n \right) =f(e)$
Thus, you have,
$\lim_{n\to\infty}\left( \frac{n}{n+1} \right)^n=1/e$
Again, since $g(x)=x^2$ is countinous you can evaluate both sides by it thus,
$g\left(\lim_{n\to\infty}\left( \frac{n}{n+1} \right)^n \right) =g(1/e)$
Thus,
$\lim_{n\to\infty}\left( \frac{n}{n+1} \right)^{2n}=1/e^2$