Can I apply L’Hopital’s rule to this:
$$ \lim_{x\to0}\frac{f(x)}{g(x)} $$
when $ \lim_{x\to0}f(x) = \infty $ and $ \lim_{x\to0}g(x) = -\infty $. Is this an indeterminate form?
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$\begingroup$Yes.
$\lim_{x\to0}\dfrac{f(x)}{g(x)}$ in which $\lim_{x\to0}f(x) = \infty,$ and $\lim_{x\to0}g(x) = -\infty$ (or vice versa)
or informally, $\lim_{x\to 0}\dfrac {f(x)}{g(x)} \to \dfrac \infty{-\infty} $, and/or such a limit that approaches $\frac {-\infty}\infty$
are indeterminate forms.
A handy list of the other indeterminate forms is given by Wikepedia. The entry itself is rather useful.
You can use l’Hospital’s rule in such each case above.
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