Uniform limit theorem and continuity at infinity
It is true. Let $\epsilon >0$. There exists $n_0$ such that $|f_n(x)-f(x)| <\epsilon $ for $n \geq n_0$ for all $x$. If $L_n =\lim_{x \to \infty} f_n(x)$ and $L =\lim_{x \to \infty} f(x)$ the we can let $x \to \infty$ in above inequality to get $|L_n-L| \leq \epsilon$ for all $ n \geq n_0$. It follows that $L_n \to L$ which proves that $$ \lim_{n \to \infty} \lim_{x \to x_0} f_n(x) = \lim_{x \to x_0} \lim_{n \to \infty} f_n(x). $$