L’Hôpital’s rule practice question 6

What do $x$ and $\sin(3/x)$ each do as $x \to \infty$? L’Hôpital’s rule works on quotients, so how could you rewrite a product?

The form is $\infty\cdot 0$. Rewrite the product as a quotient, $\dfrac{\sin(3/x)}{1/x}$, which has the form $\tfrac{0}{0}$.

Differentiate the top with the chain rule: the derivative of $\sin(3/x)$ is $\cos(3/x)\cdot\left(-\tfrac{3}{x^{2}}\right)$.