Answer
$1$
Work Step by Step
Our aim is to evaluate the limit for $\lim\limits_{x \to +\infty} \dfrac{x+\sin 2x}{x}$
But $\lim\limits_{x \to +\infty} \dfrac{x+\sin 2x}{x}=\dfrac{\infty+\infty}{\infty}$
We can see that the numerator and denominator have a limit of $\infty$, so the limit shows the indeterminate form of type $\dfrac{\infty+\infty}{\infty}$. So, we will apply L'Hopital's rule which can be defined as: $\lim\limits_{x \to a} \dfrac{P(x) }{Q(x)}=\lim\limits_{x \to a}\dfrac{P'(x)}{Q'(x)}$
where, $a$ can be any real number, infinity or negative infinity.
$\lim\limits_{x \to +\infty} (1+\dfrac{\sin 2x}{x})=\lim\limits_{x \to +\infty} (1)+\lim\limits_{x \to +\infty} \dfrac{\sin 2x}{x}\\=1+0\\=1$