# $\lim_{n \rightarrow \infty}\frac{x}{n}(1+\frac{x}{n})^{n-1} = ?$

I do not understand why this expression simplifies to

$$xe^x$$

My intuition tells me

\begin{align} \lim_{n \rightarrow \infty}\frac{x}{n}\left(1+\frac{x}{n}\right) ^{n-1} & = \lim_{n \rightarrow \infty}\frac{x}{n+x}\left(1+\frac{x}{n} \right)^{n} \\ \end{align}

where the right half of the expression goes to $$e^x$$ but I do not see why the left half is $$x$$.

May I have some help, please?

• Are you sure it is supposed to be $\frac{x}{n}\left(1+\frac{x}{n}\right)^{n-1}$ rather than just $x\left(1+\frac{x}{n}\right)^{n-1}$? The former goes to $0$, while the latter goes to $xe^{x}$. – Jake Mar 19 at 0:39
• You were totally right. It was my mistake. Thank you for pointing that out. – hyg17 Mar 19 at 2:19

You are right. The limit is $$0$$, not $$xe^{x}$$.