calculus - Proving that $f(x) = frac{x^2}{1+sin^2(1/x)},f(0)=0$ is continuous at $0$

I'm trying to prove that

$$f\left(x\right)=\begin{cases}
\frac{x^{2}}{1+\sin^{2}\left(\frac{1}{x}\right)}, & x\neq0\\
0, & x=0
\end{cases}$$

is continuous at 0.

I know that I should show that
$$\lim_{x\rightarrow0}\frac{x^{2}}{1+\sin^{2}\left(\frac{1}{x}\right)}=0$$

But I don't understand what is
$$\lim_{x\rightarrow0}\sin^{2}\left(\frac{1}{x}\right)=?$$
When $x$ goes to $0$, $\dfrac{1}{x}$ goes to infinity, but sine oscillates between $-1$ and $1$, hence I don't know what limit would be.

Could you, please, help me?

Answer

You should apply squeeze theorem here. We have that for $x\in\mathbb{R}\setminus \{0\}$, $-1 \le \sin\left(\frac{1}{x}\right) \le 1$. So we have that $0 \le \sin^2\left(\frac{1}{x}\right) \le 1$. Let's make use of this property.

$$0 \le \lim_{x\rightarrow 0} \frac{x^2}{1+\sin^2\left(\frac{1}{x}\right)} \le \lim_{x\rightarrow 0} \frac{x^2}{1+0}.$$

Applying the limit on the right, we have that the overall limit is $0$.