[Math] the limit of $x/(x+\sin x)$ as $x$ approaches infinity

Question

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 $0$.

Answer 1

Assume $x\neq 0$ and divide the given term by $x$ to get the form $\frac{1}{1+\frac{\sin(x)}{x}}$. This clearly tends to $1$ as $x\rightarrow\infty$ since $-1\le\sin(x)\le1$.