Math

L'Hôpital's Rule Calculator

Solve Indeterminate Limits with L'Hôpital's Rule

When a limit yields an indeterminate form like 0/0 or ∞/∞, L'Hôpital's rule lets you differentiate the numerator and denominator separately and re-evaluate. This calculator detects indeterminate forms and evaluates the limit numerically, helping you verify your L'Hôpital's rule solutions.

Limit Calculator Tips

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Try an Example

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Classic Sinc Function

Evaluate lim sin(x)/x as x approaches 0 - a foundational calculus limit

Key values: sin(x)/x · x -> 0 · Result: 1

Exponential Growth

Explore the limit of (1 + 1/x)^x as x approaches infinity (Euler's number)

Key values: (1+1/x)^x · x -> Infinity · Result: e

Difference Quotient

Evaluate the derivative definition limit for x² at x = 3

Key values: (x² - 9)/(x - 3) · x -> 3 · Result: 6

Documentation

This calculator is also known as L'Hôpital's Rule Calculator.

Read the complete guide

What is L'Hôpital's Rule?

L'Hôpital's rule states: if lim(x→a) f(x)/g(x) yields 0/0 or ∞/∞, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x), provided the latter limit exists.

Examples

Classic 0/0 Form: sin(x)/x

Evaluate lim(x→0) sin(x)/x.

Direct substitution gives 0/0 (indeterminate). By L'Hôpital's rule or the squeeze theorem: lim = cos(0)/1 = 1.

Key takeaway: lim(x→0) sin(x)/x = 1 is one of the most fundamental limits in calculus.

Evaluating Limits: Strategy

Follow this order when evaluating limits:

First try direct substitution — if the result is a number, that is the limit
If you get 0/0 or ∞/∞, check for algebraic simplification (factoring, rationalizing)
Apply L'Hôpital's rule if algebraic methods do not resolve the indeterminate form
Use the squeeze theorem for oscillatory functions like sin(x)/x
Verify numerically by checking f(x) for x values close to the limit point

Frequently Asked Questions about L'Hôpital's Rule Calculator

When can L'Hôpital's rule be applied?

L'Hôpital's rule applies only when the direct substitution yields 0/0 or ∞/∞. It does NOT apply to other indeterminate forms (0·∞, ∞-∞, etc.) without algebraic transformation first.

What are the common indeterminate forms?

The seven indeterminate forms are: 0/0, ∞/∞, 0·∞, ∞-∞, 0^0, 1^∞, and ∞^0. Each requires different manipulation before applying limit rules.

How many times can L'Hôpital's rule be applied?

L'Hôpital's rule can be applied repeatedly as long as the result remains in an indeterminate form. Each application differentiates the numerator and denominator once more.