calculus - What is the value for $lim_{xtoinfty} frac{sin x}{x}$?

What is the value for $\lim \limits _{x\to\infty} \frac{\sin x} x$?

I solved it by expanding $\sin x$ as

$$\sin x = x - \frac {x^3} {3!} \dotsc$$

So $\lim \limits _{x\to\infty} \frac {\sin x} x = 1 -\infty = - \infty$,

but the answer is $0$. Why? What I am doing wrong?

Answer

Yes , the answer is $0$ .

One way to see this is by using the inequality :

$$\left |\frac{\sin x}{x}\right | \leq \frac{1}{x}$$ when $x>0$ (this happens because $|\sin x\ | \leq 1$ )

When $x \to \infty$ we have $\frac{1}{x} \to 0$ so the limit must be $0$ .