TITLE: Does L-Hopital's rule fail for $\lim _{x \to \infty} \frac{x+\sin x}{x+2 \sin x}$?
QUESTION [6 upvotes]: For $$\lim _{x \to \infty} \frac{x+\sin x}{x+2 \sin x}$$ if I solve it by dividing by $x$ and I get the correct answer which is $1$ but when I apply L-Hopital rule for $\infty /\infty$ form then I get answer as limit does not exist because we cannot say anything about $$\lim _{x \to \infty} \frac{1+\cos x}{1+2 \cos x}$$
Why does L-Hopital rule fail for this case? Or is something incorrect in my approach?

REPLY [11 votes]: L'hopital says that if 
$$\lim \frac{f^{\prime}(x)}{g^{\prime }(x)}$$ exists then 
$$\lim \frac{f(x)}{g(x)}$$ exists. In your case the first limit doesnt exist, this does not contradict the rule, but you cannot draw any conclusion about the second limit.