Determine if the sequence $a_n=\frac{4^n+5n!}{n!+2^n}$ converges or diverges. If the sequence converges, find its limit. State your reasoning.

Question

I've been able to solve similar questions but the introduction of the factorial is throwing me off. I know how factorials work as in $\;8!= 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1\,,\;$ but I'm having trouble putting it into practice within the question. Thanks in advance.

$a_n=\dfrac{4^n+5n!}{n!+2^n}$