calculus - proving $limlimits_{xto infty }frac{x^2}{x^2+sin^2 x}=1$

I have to prove the following equation for homework

$$\lim_{x\to \infty }\frac{x^2}{x^2+\sin^2 x}=1$$

The proof must be done by proving that for every $e > 0$ exists a $M > 0$ so that for every $x > M$, $|f(x)-1| < e$ is true.

I can't seem to figure this one out.

I would greatly appropriate anyone who tries to help me out :) Thanks

Answer

$$\left| \frac{{{x}^{2}}}{{{x}^{2}}+{{\sin }^{2}}x}-1 \right| = \frac{{{\sin(x)}^{2}}}{{{x}^{2}}+{{\sin }^{2}}x} \leq \frac{1}{x^2}$$

Now making $\frac{1}{x^2} \leq \epsilon$ gives you the $M$....