Solve $ \lim_{x\to ∞}\frac{x+\sin(x)}{2x+\cos(x)} $ I'm having trouble finding the limit 
$$ \lim_{x\to ∞}\frac{x+\sin(x)}{2x+\cos(x)} $$ 

I've started with
$$ \lim_{x\to ∞}\frac{\frac{x}{x}+\sin(x)}{\frac{2x}{x}+\cos(x)}= \lim_{x\to ∞}\frac{1+\sin(x)}{2+\cos(x)}=\frac{1}{2}\lim_{x\to ∞}\frac{\sin(x)}{\cos(x)} $$ 

but I don't know how to go on from here or if I should've started with a different method. 

Thank you in advance!
It should be $$ \lim_{x\to ∞}\frac{x+\sin{x}}{2x+\cos{x}}=\lim_{x\rightarrow\infty}\frac{1+\frac{\sin{x}}{x}}{2+\frac{\cos{x}}{x}}=\frac{1}{2}$$
$$\frac{x+\sin x}{2x+\cos x}=\frac12+\frac{2\sin x-\cos x}{2(2x+\cos x)}\to \frac12+0$$