L'Hôpital's Rule Calculator
Enter the numerator, denominator and the point to evaluate a 0/0 or ∞/∞ limit with L'Hôpital's rule.
Use ^ for powers; enter 'infinity' for ∞.
L'Hôpital's rule evaluates a limit of a fraction that gives an indeterminate form, 0/0 or ∞/∞, by taking the derivative of the numerator and the derivative of the denominator separately, then evaluating the new limit. For (x²−1)/(x−1) at x = 1, differentiating gives 2x/1, which is 2 at x = 1.
How to Use This Calculator
Enter the numerator and denominator of your fraction as functions of x, then the value x approaches. You can enter 'infinity' for limits at infinity. The calculator evaluates the limit and shows the derivatives L'Hôpital's rule uses.
L'Hôpital's rule only applies when direct substitution gives an indeterminate form — 0/0 or ∞/∞. For other cases the limit can often be found by direct substitution instead.
How the Calculation Works
When plugging the point into a fraction gives 0/0 or ∞/∞, the ratio is indeterminate — you can't tell the limit from the form alone. L'Hôpital's rule says the limit equals the limit of the ratio of the derivatives of the top and bottom.
The calculator differentiates the numerator and denominator separately (not with the quotient rule) and evaluates the resulting limit, repeating if the new form is still indeterminate.
if f/g → 0/0 or ∞/∞, then lim f/g = lim f′/g′
Worked Example
Take the limit of (x² − 1)/(x − 1) as x → 1. Substituting gives (1 − 1)/(1 − 1) = 0/0, an indeterminate form, so L'Hôpital's rule applies.
Differentiate top and bottom: the derivative of x² − 1 is 2x, and of x − 1 is 1. The new limit is 2x/1, which at x = 1 is 2.
Assumptions and Limitations
- Applies to 0/0 and ∞/∞ indeterminate forms
- Differentiates numerator and denominator separately (not the quotient rule)
- Enter 'infinity' for limits at infinity
- Other indeterminate forms may need rewriting into a fraction first
When to Use an L'Hopital's Rule Calculator
Calculus students use this to evaluate a 0/0 or infinity/infinity limit using L'Hopital's rule. Run it to check a limit, follow the method, or handle an indeterminate form quickly. Enter the numerator, denominator, and point to get the limit.
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Open calculator →Frequently Asked Questions
When does L'Hôpital's rule apply?
Only when substituting the point into the fraction gives an indeterminate form of 0/0 or ∞/∞. If direct substitution gives a definite value, you don't need the rule.
How does L'Hôpital's rule work?
It replaces the limit of f/g with the limit of f′/g′ — the derivative of the numerator over the derivative of the denominator — which often resolves to a definite value.
Do I use the quotient rule with L'Hôpital?
No. You differentiate the numerator and the denominator separately, not the whole fraction. Using the quotient rule here is a common mistake.
What if it's still 0/0 after one step?
Apply the rule again — differentiate top and bottom once more. You can repeat as long as each new limit is still an indeterminate 0/0 or ∞/∞ form.
Can L'Hôpital handle other indeterminate forms?
Forms like 0·∞ or ∞−∞ must first be algebraically rewritten as a 0/0 or ∞/∞ fraction. Then L'Hôpital's rule can be applied to that fraction.
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