What is a limit in calculus?

A limit describes the value a function approaches as the input approaches a given point. For example, lim(x→2) x^2 = 4 because x^2 gets arbitrarily close to 4 as x approaches 2.

What is the difference between one-sided and two-sided limits?

A two-sided limit requires the function to approach the same value from both the left and the right. A one-sided limit only considers approach from one direction — left-hand (x→a⁻) or right-hand (x→a⁺).

What does DNE (Does Not Exist) mean for a limit?

DNE means the limit does not exist. This happens when the left-hand and right-hand limits are different, or when the function grows without bound (oscillates or diverges) rather than settling to a finite value.

What is L'Hopital's rule?

L'Hopital's rule states that if lim f(x)/g(x) produces 0/0 or ∞/∞, you can differentiate both numerator and denominator separately and then take the limit of that new ratio. This calculator applies it for common rational and trigonometric indeterminate forms.

Why is lim sin(x)/x = 1 as x→0?

This is the Squeeze Theorem limit. Near x = 0, sin(x) ≈ x, so sin(x)/x ≈ 1. Formally, cos(x) ≤ sin(x)/x ≤ 1 near 0, and both sides approach 1, squeezing sin(x)/x to 1.

Limit Calculator

Enter a function and a target value to evaluate one-sided or two-sided limits, with step-by-step reasoning and pre-loaded standard limits.

Evaluate a Limit

Enter the function and the value x approaches. Use inf for infinity.

f(x) =

x approaches

Direction

Two-sided Left (x→a⁻) Right (x→a⁺)

Quick Examples

Standard Limits Library

Click any entry to load it into the calculator.

Result

Enter a function above and click Evaluate Limit.

Step-by-Step Explanation

Steps will appear here after evaluation.

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Summary

This limit calculator evaluates one-sided (left-hand and right-hand) and two-sided limits of common function types directly in your browser. It handles polynomials, rational functions, trigonometric expressions, and standard indeterminate forms by applying L'Hopital's rule, factoring, and algebraic simplification. All computation is client-side — nothing is sent to a server.

How it works

Select the function type or pick a pre-loaded standard limit from the library.
Enter your function expression and the value x approaches (supports "inf" for infinity).
Choose the direction: two-sided, left-hand only, or right-hand only.
Click "Evaluate Limit" to see the result and a step-by-step explanation.
Use the pre-loaded library to instantly look up limits like sin(x)/x as x→0.

Use cases

Verify calculus homework answers for limit problems.
Look up standard limits like lim sin(x)/x as x approaches 0.
Check one-sided limits at points of discontinuity.
Explore behavior of rational functions as x approaches infinity.
Evaluate limits involving indeterminate forms (0/0, ∞/∞).
Study L'Hopital's rule applied step by step.
Quickly find limits of polynomial and root functions.
Confirm limits for exam preparation.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu