determine the limsup and the liminf as $x \to 0$ of this function?

Question

For $f(x)=\sin(x)\sin(\frac{1}{x})$ as $x\to 0 $.

limsup should be $0$ and liminf should be $1$. Is my answer correct?

Answer 1

No.

$|f(x)| \le | \sin x|$ for all $x$. Hence $f(x) \to 0$ for $x \to 0$.

Answer 2

You have here an example of the

Claim: if $\;\lim\limits_{x\to x_0}g(x)=0\;$ and $\;h(x)\;$ is a function defined in some neighborhood of $\;x_0\;$ and bounded there , then $\;\lim\limits_{x\to x_0}g(x)h(x)=0\;$

Thus, the limit of your function is zero, and then the lim sup and the lim inf. equal zero, too.