L’Hôpital’s rule practice question 17

What do $x$ and $(\log x)^{2}$ each do as $x \to 0^{+}$? The rule needs a quotient, so which factor would you rather differentiate?

The form is $0\cdot\infty$. Rewrite it as $\dfrac{(\log x)^{2}}{1/x}$, which has the form $\tfrac{\infty}{\infty}$, keeping the logarithm on top.

One application leaves $-2x\log x$, another $0\cdot\infty$ product. Rewrite it the same way and apply the rule again.