AP® Calculus AB · Unit 4 · Topic 4.7
Using L'Hospital's Rule for Indeterminate Forms
L'Hospital's Rule says that if a limit of f(x)/g(x) gives 0/0 or infinity/infinity, you can instead take the limit of f'(x)/g'(x), as long as that new limit exists.
Plug in the number and get 0/0 or infinity/infinity, and the fraction is telling you nothing yet, both pieces are collapsing or blowing up together. L'Hospital's Rule says look at how fast each piece is changing instead of what value each piece has. Replace f(x)/g(x) with f'(x)/g'(x) near that point, take the new limit, and you get the same answer the original limit was hiding. This works because near the point in question, a differentiable function behaves almost exactly like its tangent line. So f(x)/g(x) is really behaving like (tangent line of f)/(tangent line of g), and since both tangent lines pass through 0 at that point in a 0/0 case, their ratio simplifies to slope over slope, which is f'(a)/g'(a).
Sometimes one pass of the rule still leaves 0/0, so you differentiate top and bottom again. That is fine, as long as each version is still indeterminate before you apply the rule again. The rule never cares about the history of a problem, only the current form in front of you.
This sits right after indeterminate limits and linear approximation in the unit because it is really linear approximation in disguise: you are comparing two functions through their local linear behavior rather than their raw values. It connects forward to anywhere a limit shows up buried inside a derivative definition, an improper integral, or a rate comparison, since those all eventually reduce to checking whether a ratio is genuinely indeterminate before you can simplify it.
Key terms
- Indeterminate form
- An expression like 0/0 or infinity/infinity whose value is not determined until you analyze it further.
- L'Hospital's Rule
- If f(x)/g(x) is 0/0 or infinity/infinity at a point, its limit equals the limit of f'(x)/g'(x) there, if that limit exists.
- Tangent line approximation
- Near a point, a differentiable function behaves almost like the straight line tangent to it at that point.
- Determinate form
- A limit expression that already gives a clear numerical answer or a clear infinity without needing further simplification.
Watch it explained
- How it worksHow does replacing two functions by their tangent lines explain L'Hospital's Rule?
- Worked exampleFind the limit of (ex - 1 - x)/x2 as x approaches 0 with L'Hospital's Rule.
- Worked exampleFind the limit of ln(x)/(x - 1) as x approaches 1.
- Common mistakeWhy isn't L'Hospital's Rule the quotient rule in disguise?
Check yourself
End of topic.