AP® Calculus BC · Unit 4 · Topic 4.7
Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
L'Hospital's rule says that when a limit of f(x)/g(x) reads 0/0 or infinity/infinity, you can replace it with the limit of f'(x)/g'(x), since both functions are collapsing at matched rates near that point.
When a limit gives 0/0, the usual algebra tricks (factoring, rationalizing) often stall. But 0/0 secretly carries information: both the top and bottom are shrinking to zero, and what matters is how fast each one shrinks. A derivative measures exactly that rate. So instead of comparing f(x) and g(x) directly, you compare their slopes f'(x) and g'(x) at the point where everything collapses. If that new ratio settles on a number, the original limit settles on the same number.
Mechanically, you never touch the quotient rule here. You differentiate the numerator and denominator separately, keeping them as two separate functions, then take the limit of the new fraction. If the result is still 0/0 or infinity/infinity, you repeat the process again on the new ratio. This only works for these two indeterminate forms; a limit like 3/0 is not indeterminate at all, it is a vertical asymptote, and L'Hospital's rule cannot be applied there. Some limits need the rule applied two or three times in a row before a clean numerical answer appears, and occasionally differentiating never breaks the cycle, which signals the limit must be found some other way, such as rewriting the expression first.
This topic is the payoff for everything unit 4 built: it turns derivatives into a tool for evaluating limits, not just slopes and rates. It connects directly to earlier work with indeterminate forms and to later uses in improper integrals and sequences, where the same 0/0 and infinity/infinity standoffs reappear.
Key terms
- Indeterminate form
- An expression like 0/0 or infinity/infinity whose value isn't determined until you examine the functions' behavior more closely.
- L'Hospital's rule
- If f(x)/g(x) gives 0/0 or infinity/infinity at a point, that limit equals the limit of f'(x)/g'(x) at the same point.
- Vertical asymptote
- A location where a function grows without bound, often signaled by a nonzero number divided by zero, not an indeterminate form.
- Repeated differentiation
- Applying L'Hospital's rule more than once when the new ratio of derivatives is still indeterminate.
Watch it explained
- How it worksHow does swapping a ratio of functions for a ratio of derivatives preserve the limit?
- Worked exampleFind the limit as x approaches 0 of (ex - 1 - x)/x2 using L'Hospital's rule.
- Worked exampleFind the limit as x goes to infinity of x2/ex by applying the rule twice.
- Common mistakeWhy doesn't L'Hospital's rule apply to a limit that already reads 3/0?
- Going furtherWhat happens when the rule cycles forever and never resolves?
Check yourself
End of topic.