Since 2010 · Powering 2M+ tool runs every month
Since 2010
Add to Chrome

My Toolbox

Automatic Mode

No saved tools yet.

Go Premium
Related tools
Limit CalculatorSingle Variable Derivative CalculatorImproper Integral Calculator
Home Page > Math > Calculus

L'Hôpital's Rule Calculator

Evaluate qualifying 0/0 or infinity-over-infinity limits with MiniWebtool’s L'Hôpital's Rule Calculator. Enter numerator, denominator, and approach point to follow the derivative steps used to resolve the indeterminate form.

Need help using this tool? Ask AI
Gemini (copy & open)

For Gemini, paste the copied question into the chat. You can also copy it below.

View the question

3 free uses dailyNo sign-up requiredInstant Results
L'Hôpital's Rule CalculatorTry it now — free ▼
For 0/0, f/g near a tends to the ratio of slopes f′(a)/g′(a)

Enter a numerator, a denominator and the point (infinity works too) below to resolve a 0/0 or ∞/∞ limit by differentiating, with the steps.

Try:
$$\text{If } \lim_{x \to a}\frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}, \text{ then } \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$$
YOUR LIMIT
✏ Functions
x^2, sin(x), cos(x), exp(x), ln(x), sqrt(x)
→ Approach Point
Use inf for ∞, -inf for −∞, pi, e

Embed L'Hôpital's Rule Calculator Widget

About L'Hôpital's Rule Calculator

The L'Hôpital's Rule Calculator evaluates limits that result in indeterminate forms — those frustrating 0/0 or ∞/∞ cases where direct substitution fails. Named after the French mathematician Guillaume François Antoine de l'Hôpital (1661–1704), this rule transforms difficult limit problems into simpler ones by differentiating the numerator and denominator separately. This calculator automates the entire process, applying the rule iteratively with fully rendered MathJax step-by-step solutions, so you can follow every derivative and substitution.

At a Glance

L'Hôpital's Rule Calculator evaluates limits that give 0/0 or infinity/infinity by differentiating the numerator and denominator, repeating until the limit resolves. It shows every step.

L'Hôpital's Rule Calculator: features at a glance
FeatureWhat you getGood for
InputsEnter the numerator f(x), the denominator g(x) and the point x approaches, such as 0, pi or inf. Functions include sin, cos, tan, exp, ln, sqrt and powers.Limits typed in standard math notation
DirectionTake the limit from both sides, from the right (a+) or from the left (a-).One-sided limits
Indeterminate formsChecks that substitution gives 0/0 or infinity/infinity before applying the rule.Knowing when the rule applies
Repeated differentiationDifferentiates both functions and repeats until the limit resolves. For example, (tan x - x) / x3 at 0 takes 3 iterations and gives 1/3.Limits that need more than one pass
Steps and graphShows each derivative and substitution in formatted math, an iteration flow diagram and a function graph.Following the working for homework
Example presetsSix presets such as sin(x)/x and x2/ex load classic limits.Trying known examples quickly

What Is L'Hôpital's Rule?

L'Hôpital's Rule states: if \( \lim_{x \to a} f(x) = 0 \) and \( \lim_{x \to a} g(x) = 0 \) (or both approach ±∞), and if \( g'(x) \neq 0 \) near \( a \), then:

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

provided the limit on the right exists (or is ±∞). The key insight is that the rate of change of each function near the point determines how their ratio behaves.

Indeterminate Forms

0️⃣
0/0 Form
The most common case. Both numerator and denominator approach 0, so the ratio is undefined without further analysis. Example: \( \lim_{x \to 0} \frac{\sin x}{x} \)
♾
∞/∞ Form
Both functions grow without bound. The limit depends on which grows faster. Example: \( \lim_{x \to \infty} \frac{x^2}{e^x} \)
🔄
0 × ∞ Form
Not directly L'Hôpital-eligible, but can be rewritten as \( \frac{f(x)}{1/g(x)} \) to create a 0/0 or ∞/∞ form. Example: \( \lim_{x \to 0^+} x \ln x \)
⚡
∞ − ∞ Form
Combine into a single fraction first. Example: \( \lim_{x \to 0} \left(\frac{1}{\sin x} - \frac{1}{x}\right) = \lim_{x \to 0} \frac{x - \sin x}{x \sin x} \)

How to Use the L'Hôpital's Rule Calculator

  1. Enter the numerator f(x) — Type the numerator function using standard math notation. Supported functions: sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), x^n, and constants like pi and e.
  2. Enter the denominator g(x) — Type the denominator function. For example, for the limit of sin(x)/x, enter x here.
  3. Set the approach point — Enter the value x approaches. Use 0, pi, 1, etc. For infinity, enter inf. Select the direction: both sides, from the right (x → a⁺), or from the left (x → a⁻).
  4. Click Calculate — The calculator checks the indeterminate form, differentiates both functions, and repeats until the limit resolves. View every step with MathJax-rendered formulas, an iteration flow diagram, and a function graph.

Classic Examples

L'Hôpital's Rule Calculator: Classic Examples
LimitFormIterationsResult
\( \lim_{x \to 0} \frac{\sin x}{x} \)0/011
\( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \)0/021/2
\( \lim_{x \to 0} \frac{e^x - 1}{x} \)0/011
\( \lim_{x \to \infty} \frac{x^2}{e^x} \)∞/∞20
\( \lim_{x \to 1} \frac{\ln x}{x - 1} \)0/011
\( \lim_{x \to 0} \frac{\tan x - x}{x^3} \)0/031/3

When L'Hôpital's Rule Does Not Apply

Frequently Asked Questions

What is L'Hôpital's Rule?
L'Hôpital's Rule states that if lim f(x)/g(x) as x approaches a gives an indeterminate form 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided this new limit exists. It is named after the French mathematician Guillaume de l'Hôpital.
When can you use L'Hôpital's Rule?
You can use L'Hôpital's Rule only when direct substitution gives an indeterminate form of 0/0 or ∞/∞. It does not apply to forms like 0/1, 1/0, or other determinate forms. Both f(x) and g(x) must be differentiable near the point of interest.
Can L'Hôpital's Rule be applied more than once?
Yes. If after applying L'Hôpital's Rule the result is still an indeterminate form (0/0 or ∞/∞), you can apply the rule again to f'(x)/g'(x). This process can be repeated as many times as needed until the limit resolves or another method is required.
What are common mistakes with L'Hôpital's Rule?
Common mistakes include applying the rule when the form is not indeterminate, taking the derivative of the entire fraction (quotient rule) instead of differentiating numerator and denominator separately, and not checking that the conditions for the rule are satisfied at each step.
What indeterminate forms can be converted for L'Hôpital's Rule?
Forms like 0 × ∞, ∞ − ∞, 0⁰, 1^∞, and ∞⁰ can be algebraically rewritten as 0/0 or ∞/∞ fractions, making them suitable for L'Hôpital's Rule. For example, 0 × ∞ can be rewritten as f(x)/(1/g(x)).

Reference this content, page, or tool as:

"L'Hôpital's Rule Calculator" at https://MiniWebtool.com/l-h-pital-s-rule-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

Chat with your favorite AI

Gemini (copy & open)

For Gemini, paste the copied question into the chat. You can also copy it below.

View the question

Calculus:

Math SolverWhat would you like to solve?

Math Solver

Try
Browse expression examples

Choose an example to fill the input, edit its numbers, then Solve. log uses base 10; use ln for the natural logarithm.

Numbers & factors

Fractions & division

Powers, roots & logs

Algebra & equations

Calculus

Geometry & trigonometry

Top & Updated:

Simpson's Rule CalculatorCramer's Rule CalculatorHiking Pace Calculator (Naismith's Rule)View all →
Home Page > Math > Calculus > L'Hôpital's Rule Calculator