convergence of an ugly sequence

Question

Determine the convergence of the sequence $$a_n=\frac{2n+\cos (n^2)}{n+(-1)^n\sqrt{n}+\sin n}$$.

I knew that there is a short solution useing the L'hopital's rule (and it converges to 2), but the theorem is not allowed, but then I dont know how to prove it converges, somebody please help.

Answer 1

$$a_n=\dfrac{2+\dfrac{\cos n^2}{n}}{1+\dfrac{(-1)^n}{\sqrt{n}}+\dfrac{\sin n}{n}}.$$

Answer 2

$$ a_n=\frac{2n+\cos (n^2)}{n+(-1)^n\sqrt{n}+\sin n}$$

or

$$ a_n = \frac{2+ \dfrac {\cos (n^2)} n}{1+\dfrac{(-1)^n}{\sqrt{n}}+ \dfrac{\sin n}{n}} $$

Now let $n \to \infty$