Solving $ \lim_{n \to \infty} \frac{n^{10} - n!}{n^{12}}$

Question

I want to calculate this: $$ \lim_{n \to \infty} \frac{n^{10} - n!}{n^{12}} $$ I suppose it is $$ -\infty $$ since $n!$ is really huge, but how do I prove it..?

Answer 1

Hint: You have, for $n$ big enough, $$\dfrac{n!}{n^{12}} = \dfrac{n}{n} \dfrac{n-1}{n} \dfrac{n-2}{n}\dfrac{n-3}{n}\dfrac{n-4}{n}\dfrac{n-5}{n}\dfrac{n-6}{n}\dfrac{n-7}{n}\dfrac{n-8}{n}\dfrac{n-9}{n}\dfrac{n-10}{n}\dfrac{n-11}{n}\dfrac{(n-12)!}{1}$$