Can we apply L'hopital Rule when we have $\frac{-\infty}{+\infty}$?

Question

I saw in a book, it evaluated $\lim_{x\to0^+}\cfrac{(\ln x)^3}{\frac1x}$ with L'hopital Rule. the fraction is $\frac{-\infty}{+\infty}$ actually. so can I use L'hopital rule safely every time I see something like $\frac{-\infty}{+\infty}$ ?

Answer 1

Sure. That's so because, if $\lim_{x\to a}f(x)=-\infty$ and $\lim_{x\to a}g(x)=\infty$, then\begin{align}\lim_{x\to a}\frac{f(x)}{g(x)}&=-\lim_{x\to a}\frac{-f(x)}{g(x)}\\&=-\lim_{x\to a}\frac{(-f)'(x)}{g'(x)}\\&=-\lim_{x\to a}\frac{-f'(x)}{g'(x)}\\&=\lim_{x\to a}\frac{f'(x)}{g'(x)}.\end{align}

Answer 2

hint

Write

$$\frac{f(x)}{g(x)}=\frac{\frac{1}{g(x)}}{\frac{1}{f(x)}}$$

to transform $ \frac{\infty}{\infty} $ into $ \frac 00$.