limit(x->infinity) (sin(2 x))/(x+sin(x))Please show workThanks in advance
Note that for all x greater then 1
$\displaystyle x-1 \le x+\sin(x) \le x+1 \iff \frac{1}{x+1} \le \frac{1}{x+\sin(x)} \le \frac{1}{x-1}$
and note that $\displaystyle -1 \le \sin(2x) \le 1$
Putting these two facts together gives
$\displaystyle \frac{-1}{x+1} \le \frac{\sin(2x)}{x+\sin(x)} \le \frac{1}{x-1}$
Now what happens as you take the limit?