Limit Calculator
Calculate limits with step-by-step solutions. Supports one-sided limits, limits at infinity, L'Hôpital's rule, and more.
Formula:lim(x→0) f(x)
Enter Expression
f(x)
Limit Practice Problems
Test your skills with practice problems
Practice with 5 problems to test your understanding.
What is a Limit?
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a particular value. The limit of f(x) as x approaches a, written as lim(x→a) f(x), is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a.
A limit describes the value that a function approaches as the input approaches a specific value. Use our free calculator to find limits instantly with step-by-step solutions. Simply enter any function like x^2, sin(x)/x, or (1+1/x)^x and get the limit with detailed explanations.
Key Facts About Limits
- •lim(x→0) sin(x)/x = 1 (fundamental trigonometric limit)
- •lim(x→∞) (1+1/x)^x = e (definition of Euler's number)
- •lim(x→0) (e^x-1)/x = 1
- •L'Hôpital's Rule: For 0/0 or ∞/∞, take derivatives of numerator and denominator
- •One-sided limits: x→a⁺ (from right) and x→a⁻ (from left)
- •A two-sided limit exists only if both one-sided limits exist and are equal
- •If direct substitution works, the function is continuous at that point
- •Limits at infinity describe end behavior of functions
Frequently Asked Questions
What is the limit of sin(x)/x as x approaches 0?
The limit of sin(x)/x as x approaches 0 equals 1. This is one of the most important limits in calculus and can be proven using the squeeze theorem or geometric arguments.
When should I use L'Hôpital's Rule?
Use L'Hôpital's Rule when direct substitution gives an indeterminate form like 0/0 or ∞/∞. Take the derivative of both numerator and denominator, then try evaluating the limit again.
What is a one-sided limit?
A one-sided limit examines function behavior from only one direction. The left-hand limit (x→a⁻) approaches from values less than a, while the right-hand limit (x→a⁺) approaches from values greater than a.
What does it mean when a limit does not exist?
A limit does not exist (DNE) when: the left and right limits are different, the function oscillates infinitely, or the function approaches different infinities from each side.
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).