L'Hôpital's Rule

For 0/0 or ∞/∞: f/g = f'/g'.

The idea

Theorem (L'Hôpital's rule).

Let $f$ and $g$ be differentiable near the point in question, with $g'$ nonzero there. Suppose $f$ and $g$ both tend to $0$, or both tend to $\pm\infty$. If $\lim \frac{f'(x)}{g'(x)}$ exists or is infinite, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}.$

The forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ are called indeterminate because knowing the two limits separately determines nothing about the quotient: two quantities both tending to zero can have a ratio approaching any value, depending on which shrinks faster. The rule compares those speeds directly, replacing each function by its rate of change: near a point $a$ where $f$ and $g$ both vanish, each graph hugs its tangent line, so the quotient $f/g$ is carried by the ratio of the slopes $f'(a)/g'(a)$.

Three cautions. First, this is a statement about limits, not a way of differentiating a quotient: numerator and denominator are differentiated separately, which is not the quotient rule. Second, the rule applies only to the indeterminate forms; applied to any other quotient it produces a wrong answer with no warning, so check the form before differentiating. Third, the implication runs one way. If the new limit exists, the old one equals it; if the new limit fails to exist, the rule gives no information, and the original limit may exist anyway.

Ways to work on it

• Walkthrough. L'Hôpital's rule on worked 0/0 and ∞/∞ examples.
• Practice. Random L'Hôpital limits — trig, exponential, log.
• Hardest. Conceptual: when does L'Hôpital apply, what does it not claim, and how to handle other indeterminate forms.

Not sure where to start? Take the ten-question placement test.