Prove the function $f(x)=x+\frac{\sin x}{x}$ is uniformly continuous

Question

I'm asked to prove $f(x)=x+\frac{\sin x}{x}$ is uniformly continuous in $(1,\infty)$.
I extended the continuity to $[1,\infty)$ aiming for an incoming use of Cantor's theorem on a function defined over a closed interval. To do so I had to show the derivative is bounded and it didn't go well, I kept going back into the loop of $\frac{\infty}{\infty}$ limit.

Answer 1

$f(x)-x=\frac {\sin x} x$ is continuous and tends to $0$ as $x \to \infty$. This implies that $f(x)-x$ is uniformly continuous on $[1,\infty)$. [Thanks to Ennar for providing a link for this implication in the comment below]. Sum of two uniformly continuous functions is uniformly continuous, so $f(x)=[f(x)-x]+x$ is uniformly continuous.