$f_n(x) =\frac{n}{2}\left(e^{\left(\frac{1}{n}-4\right)x}-e^{\left(-\frac{1}{n}-4\right)x}\right)$ - Prove uniform convergence in $[-1,1]$

Question

$$f_n(x) =\frac{n}{2}\left(e^{\left(\frac{1}{n}-4\right)x}-e^{\left(-\frac{1}{n}-4\right)x}\right).$$ Need to prove uniform convergence in $[-1,1]$
I first calculated the P.W converge, which is: $e^{-4x}$
I then had to prove that it doesnt converge uniformly in $(-\infty, -1]$.
Done it, but now I have to prove in: $[-1,1]$ ( then at $[1, \infty)$ - but I think I will manage it, but first I need to prove the $[-1,1]$

Answer 1

If $x\neq 0$, then we have $$ f_n (x) = xe^{ - 4x} \frac{{\sinh (x/n)}}{{x/n}} \to xe^{ - 4x} $$ uniformly on compact subsets of $\mathbb R$, since if $|x|\leq M$ then $$ \left| {\frac{{\sinh (x/n)}}{{x/n}}} \right| =\frac{{\sinh (x/n)}}{{x/n}}\le \frac{{\sinh (M/n)}}{{M/n}} \to 1. $$ The case $x=0$ is obvious.