TITLE: What is the limit of $x/(x+\sin x)$ as $x$ approaches infinity?
QUESTION [15 upvotes]: I am trying to determine
$$\lim_{x \to \infty} \frac{x}{x+ \sin x} $$
I can't use here the remarkable limit (I don't know if I translated that correctly) $ \lim_{x\to 0} \frac{\sin x}{x}=1$ because $x$ approaches infinity, not $0$.

REPLY [5 votes]: Think of $\displaystyle\frac{\text{1 trillion}}{(\text{1 trillion}) + \sin(\text{1 trillion})} = \frac{\text{1 trillion}}{(\text{1 trillion}) + (\text{a number between $1$ and $-1$})}$.
If you understand that, then you will see how that leads to the answer.
To do it in a more precise way, squeeze:
$$
\frac x {x+1} \le \frac x {x+\sin x} \le \frac x {x-1}
$$
Squeezing is usually something to be considered when dealing with sines.