calculus - What is the solution to the following limit?

How to solve the following limit problem?

$$\lim_{x\to\infty} \frac{3x^2+\sin x}{x^2+(\sin x)^2}$$ . I have tried the following and arrived at a solution. But not sure if it is correct.

Applying L'Hospital's rule taking derivative thrice I get the following expression:

$$\lim_{x\to\infty} \frac{\cos x}{8\cos 2x} = 0$$