Math · Calculus I · Concept
L’Hôpital’s rule for indeterminate limits
L’Hôpital’s rule says that if f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞ at a, its limit equals the limit of f′(x)/g′(x), provided that limit exists. Differentiate the top and bottom separately, not with the quotient rule, and check the form before every use.
When the rule applies
Substitute first. If the result is a number, that is the limit and the rule is not needed. Only the forms 0/0 and ∞/∞ qualify directly; other forms, such as 0 · ∞ or ∞ − ∞, must be rewritten as a quotient first.
Differentiate the top and bottom separately
The rule uses f′ over g′, not the derivative of the quotient f/g. Differentiate the numerator and the denominator on their own, then take the new limit.
Use it again if needed
If the new limit is still 0/0 or ∞/∞, apply the rule again. Simplify between steps: cancelling a common factor often ends the problem sooner.
Why it works
Near a, a differentiable function is close to its tangent line. If f(a) = g(a) = 0, then f(x) ≈ f′(a)(x − a) and g(x) ≈ g′(a)(x − a), so their quotient is close to f′(a)/g′(a).
Other indeterminate forms
Rewrite 0 · ∞ as a quotient: x ln x = (ln x)/(1/x), which is ∞/∞ as x → 0⁺. For powers such as 1^∞, take the natural logarithm first, find the limit of the logarithm, then exponentiate.
Comparing growth rates
Used repeatedly, the rule shows that exponentials outgrow powers and powers outgrow logarithms: as x → ∞, xⁿ/eˣ → 0 for every power n, and (ln x)/xᵖ → 0 for every p > 0.
Common mistakes
- Applying the rule when the form is not 0/0 or ∞/∞: (x + 1)/(x + 2) at x = 0 is simply 1/2, but differentiating would wrongly give 1.
- Using the quotient rule on f/g instead of differentiating the top and bottom separately.
- Applying the rule a second time without checking the form again.
- Applying the rule to 0 · ∞ or 1^∞ without rewriting it first.
Key terms
- L’Hôpital’s rule
- For a limit of the form 0/0 or ∞/∞, the limit of f(x)/g(x) equals the limit of f′(x)/g′(x) when that limit exists and f and g are differentiable nearby. It doesn’t apply to other quotients.
- Indeterminate form
- A pattern such as 0/0 or ∞ − ∞ that substituting gives, which doesn’t decide the limit on its own. Different limits can give the same pattern, so more work is needed, such as factoring or L’Hôpital’s rule.
- Limit
- The value f(x) approaches as x gets closer and closer to a point a, not counting x = a itself. A limit can exist even when f(a) is undefined or has a different value.
- Derivative
- The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
Work through an example
Evaluate the limit of (1 − cos x)/x² as x → 0.
Apply L’Hôpital’s rule twice →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Check the limit in Limits & one-sided behavior Check each step in Math Open worked example on a board Limit of a quotient in Math ReferenceYour existing work stays on this device. Examples open as editable copies.