Convergence of sequence of function $\frac{|x|^n}{n+|x|^n}$

Question

When does this sequence of functions converges uniformly: $$f_n(x)=\frac{|x|^n}{n+|x|^n}$$

As I observed, this sequence converges to $0$ pointwise in $[-1,1]$ but I am unable to tackle the case for uniform convergence.

Answer 1

Perhaps you could note that if $|x| \le 1$ then $|f_n(x)| \le {1 \over n}$?

If $|x| >1$ then $f_n(x) = {1 \over 1+{n \over |x|^n}}$, so $f_n(x) \to 1$, but the convergence is not uniform.

Note that $f_n({\sqrt[n]{n}}) = {1 \over 2}$ for all $n$.