This is how I would evaluate $\lim\limits_{x \to \infty} \dfrac{x+ \sin x}{x+ 2 \sin x}$

$=\lim\limits_{x \to \infty} \dfrac{x \left( 1+ \frac{\sin x}{x} \right)}{x \left(1+ 2 \cdot \frac{ \sin x}{x} \right)}$

$= \dfrac{1+0}{1+2 \cdot 0} = 1$

But now applying L'hopitals Rule, I get

$\lim\limits_{x \to \infty} \dfrac{1+ \cos x}{1+ 2 \cos x}$

Since $\cos x $ just oscillates between $[-1,1]$ I think we can conclude the limit doesn't exist.

What is going on here?