Proving $\lim_{x\to\infty} \frac{x}{x+\sin{x}}=1$ using limit definition

Question

So I need to prove that $\lim_{x\to\infty} \frac{x}{x+\sin{x}}=1$ using the limit definition. Given $\varepsilon>0$ I'm trying to find an $M$ such that for all $x>M$, $$\left | \frac{x}{x+\sin{x}}-1 \right |=\left | \frac{\sin x}{x+\sin{x}} \right | < \varepsilon$$

I tried playing with the triangle inequality and with the inequality $|\sin x| \leq |x|, \forall x \in \mathbb{R}$ but with no luck (can't do anything with the denominator). Any suggestions?

Answer 1

Hint: for $x /ge 1$, $x + \sin x \ge x - 1$. And $|\sin x| \le 1$.