Find $\lim\limits_{x\to \infty} \frac{x\sin x}{1+x^2}$ 
$$\lim_{x\to \infty} \frac{x\sin x}{1+x^2}$$

Using L'hopital I get: 
$$\lim_{x\to \infty} \frac{x\cos x + \sin x}{2x}=\lim_{x \to \infty}\frac{\cos x}{2}$$
However, how is it possible to evaluate this limit?

Don't think L'Hospital applies here - the top limit does not exist. Hence, you are forced resort to a more fundamental approach like pointed out in other answers
