L’Hôpital’s rule practice question 16

Look at each piece near 0: $x^{2}$, $\sin(1/x)$ and $\sin x$. Which stay bounded, which tend to 0, and what does that suggest about the whole quotient?

The form is $\tfrac{0}{0}$, since $\left|x^{2}\sin(1/x)\right| \le x^{2}$. Try the rule: the derivative of the top is $2x\sin(1/x) - \cos(1/x)$. Does the new quotient have a limit?

When the quotient of derivatives has no limit, the rule says nothing. Split the original instead: $\dfrac{x}{\sin x}\cdot x\sin(1/x)$, and use $\left|x\sin(1/x)\right| \le |x|$.