45: Since cosine is a nice function, it can be pulled out of the limit:
While the indeterminate ratio in the limit looks simple, the l'Hospital rule is not going to help this time: How do you take derivative of a factorial? In fact, it even takes a lot of work just to turn n! into a function defined on all positive real numbers (see the Gamma function).
What to do? In fact, we know that this ratio converges to zero, and if you are allowed to use the scale of powers as an argument, you are done (just do not forget to put this zero into the cosine).
What if somebody wants you to provide some argument that the ratio converges to zero? The only hope really comes from some algebra. Try to write what the factorial means and look for some estimate that would allow you to pass to something simpler. After all, you want to show that the ratio goes to zero, so if you find an upper estimate that also goes to zero, you are done by the absolute value version of the Squeeze theorem