L'Hospital's Rule and the Indeterminate Forms

The rule, stated with its hypotheses

Suppose $f$ and $g$ are differentiable near $a$ with $g'\ne0$ near $a$, and suppose $\lim_{x\to a}\frac{f(x)}{g(x)}$ has the indeterminate form $\frac00$ or $\frac{\infty}{\infty}$. If $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ exists (finite or infinite), then

Two hypotheses do real work here. First, the form must be checked every single time, including before each repeat application. Second, the implication runs one way only: if the new limit fails to exist, the rule tells you nothing at all about the original — the original may still converge.

The other five forms

Only $\frac00$ and $\frac{\infty}{\infty}$ may be differentiated directly. The forms $0\cdot\infty$, $\infty-\infty$, $1^{\infty}$, $0^{0}$ and $\infty^{0}$ must first be rewritten:

• $0\cdot\infty$: move one factor into the denominator as its reciprocal.
• $\infty-\infty$: combine over a common denominator, or multiply by a conjugate.
• $1^{\infty}$, $0^{0}$, $\infty^{0}$: set $y$ equal to the expression, take $\ln$, find $\lim\ln y=L$, and answer $e^{L}$.

Note that $1^{\infty}$ is genuinely indeterminate: $\left(1+\frac1x\right)^{x}\to e$ while $\left(1+\frac{1}{x^{2}}\right)^{x}\to1$, and both have that same form.