Generally true or false? $\lim_{n\to\infty} f(g(n)) \ne f(\lim_{n\to\infty} g(n)) $

Generally true or false? $\lim_{n\to\infty} f(g(n)) \ne
f(\lim_{n\to\infty} g(n)) $

I came across a proof in a probability textbook proof for the proposition
$$\lim_{n\to\infty} P(E_n) = P(\lim_{n\to\infty} E_n) $$
which prompts me to ask, (why) is the following generally true or false?
$$\lim_{n\to\infty} f(g(n)) = f(\lim_{n\to\infty} g(n)) $$
Although the mere presence of the proof already implies "false", could
someone explain why the $\lim$ is not necessarily associative with respect
to functions?