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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?

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    $\begingroup$ 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 $xe^{x}$. $\endgroup$ – Jake Mar 19 at 0:39
  • $\begingroup$ You were totally right. It was my mistake. Thank you for pointing that out. $\endgroup$ – hyg17 Mar 19 at 2:19
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You are right. The limit is $0$, not $xe^{x}$.

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