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Calculator Answers

L'Hopital Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An L'Hopital's rule calculator is a calculus utility that evaluates limits of indeterminate forms, specifically zero over zero or infinity over infinity. The calculator verifies if the limit matches an indeterminate form, then applies L'Hopital's rule by taking the derivatives of the numerator and denominator independently. It recursively applies this differentiation step until a determinate limit value can be calculated. It outputs the derivatives and the final limit value. Calculus students use this tool to understand limit differentiation steps.

Enter polynomial coefficients for the numerator and denominator, and the value x = a. The calculator evaluates the original ratio; if it is 0/0, it differentiates both polynomials and evaluates the new ratio.

Quick Answer

Solve indeterminate limit forms using L'Hopital's rule. Enter your fraction function and target value to see derivatives and step-by-step solutions.

Polynomial coefficients

Enter coefficients in descending order. For example, x − 1 is 1, -1; x² − 4 is 1, 0, -4.

e.g. 1, -1

e.g. 1, -1

e.g. 1

Scope

This calculator applies L'Hôpital's rule once to a polynomial-over-polynomial limit at a finite x = a. If direct substitution gives 0/0, the calculator differentiates both sides and evaluates the new ratio.

Out of scope: ∞/∞ forms, exponentials and logarithms, products requiring rewriting, and multiple L'Hôpital iterations. For ∞/∞ on rationals, use the limits-at-infinity calculator.

L'Hôpital

Limit (after L'Hôpital)

1

Indeterminate 0/0 at x = 1; differentiated and re-evaluated to 1.

Numeratorx - 1
Denominatorx - 1
f(a)0 / 0
f'(numerator)1
f'(denominator)1
After L'Hopital1 / 1

L'Hôpital's rule applies only when direct substitution gives an indeterminate form like 0/0 or ∞/∞. When it does, you replace the limit of f/g with the limit of f'/g' (the derivative ratio).

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Examples

lim (x − 1) / (x − 1) at x = 1

0/0; after rule: 1/1 = 1

lim (x² − 1) / (x − 1) at x = 1

0/0; after rule: 2x/1 at 1 = 2

lim (x² − 4) / (x² − x − 2) at x = 2

0/0; after rule: 4/3

lim (3x − 6) / (x² − 4) at x = 2

0/0; after rule: 3/4

How it works

L'Hopital's rule turns a hard 0/0 limit into a manageable derivative ratio. The polynomial derivative uses the power rule, applied term-by-term to the coefficient list.

Rule · lim f/g at a = lim f'/g' at a (when 0/0)

Power rule · d/dx (xⁿ) = n · xⁿ⁻¹

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Frequently asked questions

If lim f(x)/g(x) gives an indeterminate form 0/0 or infinity/infinity at x = a, then under mild conditions the limit equals lim f'(x)/g'(x). The rule replaces a hard limit with an easier one.

Only on indeterminate forms 0/0 or infinity/infinity. If direct substitution gives a defined value (like 3/2), L'Hopital does not apply; the limit is simply that value.

Apply the rule again. The calculator applies the rule once. If the result is still 0/0, it flags the case and asks you to apply the rule again by hand or with a CAS.

Mathematically yes, but this calculator handles polynomial-over-polynomial inputs only. For trig, exp, and log limits, use a CAS or apply the rule by hand using the standard derivative rules.

Coefficient input is unambiguous and never breaks. A general symbolic parser is fragile and easy to get wrong. List the polynomial coefficients in descending order and the calculator handles the rest.

The limit calculator covers polynomial and rational expressions with direct substitution and one round of (x − a) cancellation. The L'Hopital calculator is specifically for 0/0 indeterminate forms where you want to see the derivative ratio step.