Answer
The limit does not exist.
Work Step by Step
Our aim is to evaluate the limit for $\lim\limits_{x \to +\infty} \dfrac{x(2+\sin 2x)}{x+1}$
But $\lim\limits_{x \to +\infty} \dfrac{x(2+\sin 2x)}{x+1}$ does not show any indeterminate form . So, we will not apply the L'Hospital's rule .
$\lim\limits_{x \to +\infty} \dfrac{x(2+\sin 2x)}{x+1}=\lim\limits_{x \to +\infty} \dfrac{2+\sin 2x)}{1+\dfrac{1}{x}}$
We can see that there is no finite value . So, the limit does not exist.
