calculus - How to solve limits with Taylor expansion?

I'm in trouble with Taylor series..... how can I solve limits without Bernoulli-de L'Hôpital method?? For example, $$\lim_{x \to +\infty} \frac{x-\sin{x}}{2x+\sin{x}}.$$

The answer, if I'm not wrong is $\frac{1}{2}$... How, can I show that?
Thanks.