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L'Hôpital's Rule Calculator

Evaluate indeterminate limits with step-by-step calculus

lim as x →

Example Problems:

Classic: sin(x)/x
lim (x→0) sin(x)/x = 1
Polynomial: (x²-1)/(x-1)
lim (x→1) (x²-1)/(x-1) = 2
Exponential: (e^x-1)/x
lim (x→0) (e^x-1)/x = 1
Logarithmic: ln(x)/(x-1)
lim (x→1) ln(x)/(x-1) = 1
Trigonometric: (1-cos(x))/x²
lim (x→0) (1-cos(x))/x² = 0.5

About L'Hôpital's Rule

  • When to use: When direct substitution gives 0/0 or ∞/∞ (indeterminate forms)
  • The rule: If lim f(x)/g(x) is indeterminate, then lim f(x)/g(x) = lim f'(x)/g'(x)
  • Requirements: Both f and g must be differentiable near the limit point
  • Multiple applications: Can be applied repeatedly if result is still indeterminate
  • Syntax tips: Use ^ for exponents (x^2), * for multiplication (2*x), standard trig functions (sin, cos, tan)
"Per differentiam ad veritatem" — Through the derivative to truth.

L'Hôpital's Rule Calculator: The Ultimate Limit Solver

Welcome to the most powerful L'Hôpital's Rule Calculator available online. This free, instant tool evaluates indeterminate limits of the form 0/0 or ∞/∞ using calculus derivatives, showing every step from substitution to final answer. Whether you're a high school student, college learner, or professional mathematician, our L'Hôpital's Rule Calculator makes complex limit problems simple and educational.

How to Use the L'Hôpital's Rule Calculator

Our L'Hôpital's Rule Calculator is designed for speed and clarity:

  1. Enter Numerator: Type f(x) like sin(x) or x^2-1
  2. Enter Denominator: Type g(x) like x or x-1
  3. Set Limit Point: Use 0, 1, or infinity
  4. Click Calculate: Get full step-by-step solution

Solve 0/0 Forms with L'Hôpital's Rule Calculator

The classic case:

lim (x→0) sin(x)/x

Step 1: Direct substitution → 0/0 → indeterminate

f'(x) = cos(x), g'(x) = 1
lim (x→0) cos(x)/1 = 1

Our L'Hôpital's Rule Calculator automates this entire process.

Solve ∞/∞ Forms with L'Hôpital's Rule Calculator

For growing functions:

lim (x→∞) (e^x)/(x^2)

Apply once: e^x / 2x → still ∞/∞

Apply again: e^x / 2 → ∞

The L'Hôpital's Rule Calculator detects when multiple applications are needed.

Step-by-Step Solution Process

Every calculation includes:

Supported Functions in L'Hôpital's Rule Calculator

Advantages of Using L'Hôpital's Rule Calculator

  • Instant Results: No manual differentiation
  • Step-by-Step: Perfect for learning
  • Error Detection: Identifies non-indeterminate cases
  • Global Access: Works worldwide, any device
  • Educational: Builds calculus intuition

    • Applications of L'Hôpital's Rule

    • Engineering: Growth rates, stability analysis
    • Physics: Velocity, acceleration limits
    • Economics: Marginal rates, elasticity
    • Computer Science: Algorithm analysis
    • Medicine: Drug decay models

    Frequently Asked Questions

    What is L'Hôpital's Rule?

    L'Hôpital's Rule states that if lim f(x)/g(x) is 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided the latter exists. Our L'Hôpital's Rule Calculator applies this automatically.

    When can I use L'Hôpital's Rule?

    Only for 0/0 or ∞/∞ forms. The L'Hôpital's Rule Calculator checks this and warns if not applicable.

    Can it handle multiple applications?

    Yes! The L'Hôpital's Rule Calculator will reapply the rule if the result is still indeterminate.

    What if the limit is at infinity?

    Type infinity. The L'Hôpital's Rule Calculator uses large values for accurate evaluation.

    Master Indeterminate Limits Today

    The most advanced L'Hôpital's Rule Calculator for students and professionals worldwide — completely free.