TITLE: Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?
QUESTION [126 upvotes]: $$\lim_{x \to \infty} \frac{x}{x+\sin(x)}$$

This is of the indeterminate form of type $\frac{\infty}{\infty}$, so we can apply l'Hopital's rule:
$$\lim_{x\to\infty}\frac{x}{x+\sin(x)}=\lim_{x\to\infty}\frac{(x)'}{(x+\sin(x))'}=\lim_{x\to\infty}\frac{1}{1+\cos(x)}$$
This limit doesn't exist, but the initial limit clearly approaches $1$. Where am I wrong?

REPLY [8 votes]: There is another useful rule, which I don't seem to have seen written down explicitly: 

Let $f, g, r$ and $s$ be functions such that $g\to\infty$ and $r, s$ are bounded.
  Then the limit of $\dfrac{f}{g}$ and the limit of $\dfrac{f + r}{g + s}$ gives the same result. 

Applied here, since $\sin x$ is bounded, the limit is the same as the limit of $\dfrac{x}{x}$.