Limit Calculator

What Is a Limit?

A limit describes the value a function approaches as the input approaches a specific point. The limit of f(x) as x approaches 2 asks: what value does f(x) get closer and closer to as x gets closer and closer to 2? Limits are the foundational concept of calculus. Derivatives are defined as limits. Integrals are defined as limits. Continuity is defined using limits. Without limits, calculus would not exist. Enter any function and point in the calculator above to evaluate limits instantly with step-by-step work.

How to Evaluate Limits?

The simplest method is direct substitution: plug the value into the function. The limit of (x² + 3) as x approaches 2 is 4 + 3 = 7. Direct substitution works whenever the function is continuous at the target point. When substitution produces an indeterminate form like 0/0, algebraic manipulation is needed. For the limit of (x² - 4)/(x - 2) as x approaches 2: direct substitution gives 0/0. Factor the numerator: (x+2)(x-2)/(x-2) = x+2. Now substitute: 2+2 = 4. The limit is 4 even though the original function is undefined at x = 2.

Indeterminate Forms

Seven indeterminate forms require special techniques: 0/0, infinity/infinity, 0 times infinity, infinity minus infinity, 0^0, 1^infinity, and infinity^0. The most common, 0/0, usually resolves through factoring, rationalizing, or L'Hopital's rule. The form infinity/infinity also responds to L'Hopital's rule. The expression 0/0 is indeterminate because different functions approaching 0/0 can have any limit: (2x)/(x) approaches 2, (x²)/(x) approaches 0, and (x)/(x²) approaches infinity, all as x approaches 0. The indeterminate form tells you that more analysis is needed, not that no limit exists.

L'Hopital's Rule

When a limit produces 0/0 or infinity/infinity, L'Hopital's rule says: take the derivative of the numerator and denominator separately, then re-evaluate the limit. For the limit of sin(x)/x as x approaches 0: direct substitution gives 0/0. Apply L'Hopital: derivative of sin(x) is cos(x), derivative of x is 1. The limit of cos(x)/1 as x approaches 0 is cos(0) = 1. L'Hopital's rule can be applied repeatedly if the result is still indeterminate. It is one of the most powerful techniques in calculus for resolving difficult limits.

One-Sided Limits

A one-sided limit approaches from only one direction. The left-hand limit (x approaches a from below, written a⁻) and the right-hand limit (x approaches a from above, written a⁺) may differ. For f(x) = 1/x as x approaches 0: from the right, the limit is positive infinity. From the left, the limit is negative infinity. Since the one-sided limits disagree, the two-sided limit does not exist. Jump discontinuities (like in piecewise functions and step functions) also have different one-sided limits. A two-sided limit exists only when both one-sided limits exist and are equal.

Limits at Infinity

Limits at infinity describe function behavior as x grows without bound. The limit of 1/x as x approaches infinity is 0 (the function gets closer to zero as x increases). For rational functions, compare the degrees of numerator and denominator. Same degree: the limit is the ratio of leading coefficients. Higher numerator degree: the limit is infinity. Higher denominator degree: the limit is 0. For (3x² + x)/(5x² - 2) as x approaches infinity: same degree (2), so the limit is 3/5. These horizontal asymptote rules quickly determine end behavior without detailed calculation.

Continuity and Limits

A function is continuous at a point if three conditions hold: the function is defined at that point, the limit exists at that point, and the limit equals the function value. Polynomials are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero. Trigonometric functions are continuous on their domains. Understanding continuity through limits explains why some functions have holes, jumps, or vertical asymptotes in their graphs, and why the intermediate value theorem guarantees solutions to certain equations.

Squeeze Theorem

The squeeze theorem (or sandwich theorem) evaluates limits by trapping a function between two others whose limits are known and equal. If g(x) is less than or equal to f(x) is less than or equal to h(x) near a point, and the limits of g and h both equal L at that point, then the limit of f also equals L. The classic application proves that the limit of sin(x)/x as x approaches 0 equals 1: since cos(x) is less than or equal to sin(x)/x is less than or equal to 1 near zero, and both bounds approach 1, sin(x)/x must also approach 1. The squeeze theorem handles cases where direct evaluation and algebraic manipulation fail.

Frequently asked questions