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L'Hopital's Rule Calculator

Apply L'Hopital's Rule to evaluate limits of indeterminate forms. Enter the coefficients of polynomial numerator and denominator to compute limits of 0/0 or infinity/infinity forms by differentiating the top and bottom, a key technique in calculus.

Reviewed by Christopher FloiedPublished Updated

This free online l'hopital's rule calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient of the highest power in the numerator

Minimum: 1

Exponent of x in the numerator (for ax^m term)

Coefficient of the highest power in the denominator

Minimum: 1

Exponent of x in the denominator (for bx^n term)

Results

Limit (after L'Hopital's Rule)

0.666667

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the L'Hopital's Rule Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The L'Hopital's Rule Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the L'Hopital's Rule Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About L'Hopital's Rule Calculator

The L'Hopital's Rule Calculator evaluates limits of indeterminate forms involving polynomial functions. When direct substitution yields 0/0 or infinity/infinity, L'Hopital's Rule allows you to differentiate the numerator and denominator separately and re-evaluate the limit. This calculator handles the common case of comparing leading polynomial terms ax^m and bx^n, determining whether the limit is finite, zero, or infinite based on the relative degrees. L'Hopital's Rule is named after the French mathematician Guillaume de L'Hopital, who published it in 1696 (though it was actually discovered by Johann Bernoulli). It is one of the most useful techniques in calculus for evaluating difficult limits.

The Math Behind It

L'Hopital's Rule states: if lim(x->c) f(x)/g(x) gives 0/0 or infinity/infinity, and if lim(x->c) f'(x)/g'(x) exists, then lim f(x)/g(x) = lim f'(x)/g'(x). The rule can be applied repeatedly if the new limit is again indeterminate. The proof relies on the Cauchy Mean Value Theorem, which generalizes the standard Mean Value Theorem. Given that f and g are continuous on [a,b], differentiable on (a,b), and g'(x) is nonzero on (a,b), there exists c in (a,b) such that f'(c)/g'(c) = [f(b)-f(a)]/[g(b)-g(a)]. Applying this with careful limit arguments yields L'Hopital's Rule. For polynomial expressions, L'Hopital's Rule illuminates the comparison of growth rates. If the numerator has degree m and denominator degree n: when m = n, the limit is the ratio of leading coefficients times m/n; when m < n, the numerator grows slower, giving limit 0; when m > n, the numerator dominates, giving limit infinity. After one application of L'Hopital's Rule, the exponents each decrease by 1, so repeated application eventually reduces to a determinate form. L'Hopital published the rule in his 1696 textbook, the first calculus textbook ever written. However, correspondence discovered later showed that Johann Bernoulli had taught L'Hopital the rule in exchange for a salary. Despite this controversy, the name has stuck. The rule extends to other indeterminate forms (0*infinity, infinity-infinity, 0^0, 1^infinity, infinity^0) after algebraic manipulation to convert them to 0/0 or infinity/infinity form.

Formula Reference

L'Hopital's Rule

lim [f(x)/g(x)] = lim [f'(x)/g'(x)]

Variables: Applicable when lim f(x)/g(x) yields 0/0 or inf/inf

For Leading Terms ax^m / bx^n as x -> inf

Limit = (a*m)/(b*n) if m=n; 0 if m<n; inf if m>n

Variables: a,b = leading coefficients; m,n = exponents

Worked Examples

Example 1: Limit of 2x^2 / 3x^2 as x -> infinity

Evaluate lim(x->inf) 2x^2 / 3x^2.

Step 1:Direct substitution gives infinity/infinity (indeterminate)
Step 2:Apply L'Hopital's: differentiate top and bottom: 4x / 6x
Step 3:Still infinity/infinity, apply again: 4/6 = 2/3
Step 4:Since m = n = 2, we get (a1*m)/(b1*n) = (2*2)/(3*2) = 4/6 = 2/3

The limit is 2/3.

Example 2: Limit of x^2 / x^3 as x -> infinity

Evaluate lim(x->inf) x^2 / x^3.

Step 1:This is infinity/infinity
Step 2:Since numerator degree (2) < denominator degree (3), the limit is 0
Step 3:Verification: L'Hopital's gives 2x / 3x^2 = 2/(3x) -> 0

The limit is 0.

Common Mistakes & Tips

  • !Applying L'Hopital's Rule when the limit is NOT indeterminate. If direct substitution gives a determinate value like 5/3, do not differentiate. L'Hopital's Rule only applies to 0/0 or infinity/infinity forms.
  • !Differentiating the quotient f/g using the quotient rule instead of differentiating f and g separately. L'Hopital's Rule says to take f'/g', not (f/g)'.
  • !Applying the rule when g'(x) = 0 at the limit point. The rule requires that the limit of f'/g' exists; if g' is also zero, you need to apply the rule again.
  • !Forgetting that the rule may need to be applied multiple times. If the first application still gives an indeterminate form, apply it again.

Related Concepts

Limit

L'Hopital's Rule is a technique for evaluating limits when direct substitution fails due to indeterminate forms.

Derivative

L'Hopital's Rule uses derivatives of the numerator and denominator to resolve indeterminate limits.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Limit Calculator

Maclaurin Series Calculator

Frequently Asked Questions

What are the indeterminate forms where L'Hopital's Rule applies?

L'Hopital's Rule directly applies to 0/0 and infinity/infinity. Other indeterminate forms like 0*infinity, infinity-infinity, 0^0, 1^infinity, and infinity^0 can be converted to 0/0 or infinity/infinity through algebraic manipulation before applying the rule.

Can L'Hopital's Rule fail?

Yes. If the limit of f'/g' does not exist (for example, it oscillates), then L'Hopital's Rule gives no conclusion. The original limit might still exist; you would need a different technique to find it.

Who really discovered L'Hopital's Rule?

Johann Bernoulli discovered the rule and shared it with Guillaume de L'Hopital as part of a paid arrangement. L'Hopital published it in his 1696 calculus textbook. While the attribution is historically inaccurate, the name L'Hopital's Rule has become standard in mathematics.

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