Does l'Hopital rule work for -inf/inf?

Question

If you have an indeterminate form:

1. $\frac{-\infty}\infty$
2. $\frac\infty{-\infty}$
3. $\frac{-\infty}{-\infty}$

does l'Hôpital's rule also apply?

Answer 1

L'Hopital can be used in all cases of $\infty$ in the denominator; it doesn't matter what the numerator is doing. (Compare with Stolz-Cesaro.)

Answer 2

Yes, but you should first make sure whether or not you really need L'Hopital's rule! If you apply it to stuff which doesn't require you to, weird things will happen.

Take for example the following limit:

$\displaystyle\lim_{x \to 0^+}x \ln x = \lim_{x \to 0^+}\dfrac{\ln x}{1/x}$.

Blindly applying L'Hopital's Rule repeatedly gives: $\displaystyle\lim_{x \to 0^+}x \ln x = \lim_{x \to 0^+}\dfrac{\ln x}{1/x} = \lim_{x \to 0^+}\dfrac{1/x}{-1/x^2} = \lim_{x \to 0^+}\dfrac{-1/x^2}{2/x^3} = \lim_{x \to 0^+}\dfrac{2/x^3}{-6/x^4} = \cdots$.

But it we stop after applying L'Hopital's Rule once and simplify stuff, we get: $\displaystyle\lim_{x \to 0^+}x \ln x = \lim_{x \to 0^+}\dfrac{\ln x}{1/x} = \lim_{x \to 0^+}\dfrac{1/x}{-1/x^2} = \lim_{x \to 0^+} -x = 0$.

Answer 3

Yes, it does apply,

Consider taking limits of these, by using l.Hopital

$1$) $\lim_ {x- \to \infty} \dfrac{-x}{x}$

$2$) $\lim_ {x- \to \infty} \dfrac{x}{-x}$

$3$) $\lim_ {x- \to \infty} \dfrac{-x}{-x}$