L’Hôpital’s rule practice question 10

As $x \to 0^{+}$, what do $x^{2}$ and $\log x$ each do? To use l’Hôpital’s rule you need a quotient: which factor is easier to keep on top?

The form is $0\cdot(-\infty)$, which is indeterminate but not a quotient. Rewrite as $\dfrac{\log x}{1/x^{2}}$, which has the form $\dfrac{-\infty}{\infty}$.

Now apply l’Hôpital: $\dfrac{1/x}{-2/x^{3}} = -\dfrac{x^{2}}{2}$.