Prove the limit

Question

Assume that $x_n$ is a sequence such that $x_n$ converge to a constant $x$. I would like to prove that

$\left( 1+ x_n/n \right)^n \rightarrow e^x$

Obvious if $x_n = x$ the result would be given. The extra sequence complicates things and I would like a hint or solution on how to proof this.

Answer 1

If you know/accept that the result holds for $x_n=x$ then the proof is easily completed by the following lemma of Thomas Andrews :

Lemma: If $n(a_n-1)\to 0$ then $a_n^n\to 1$.

Now let $$a_n=\dfrac{1+\dfrac{x_n}{n}}{1+\dfrac{x}{n}}$$ and check that $n(a_n-1)\to 0$.