How to Calculate Limits

Limit Calculator (Numerical Approximation)

Use "x" as the variable. Supported: +, -, *, /, ^, sin(), cos(), tan(), sqrt(), log(), exp().

How to Calculate Limits: A Complete Guide

In calculus, a limit describes the behavior of a function as its input variable approaches a specific value. Limits are the foundation of derivatives, integrals, and continuity.

1. Direct Substitution

The simplest way to calculate a limit is to plug the target value into the function. If the function is defined and continuous at that point, the result is the limit.

Example: lim (x -> 3) of (2x + 5)
Calculation: 2(3) + 5 = 11.

2. Factoring and Simplification

Often, direct substitution leads to an indeterminate form like 0/0. In these cases, you can factor the numerator and denominator to cancel out the problematic term.

Example: lim (x -> 1) of (x² – 1) / (x – 1)
Factor: ((x – 1)(x + 1)) / (x – 1)
Simplify: lim (x -> 1) of (x + 1) = 2.

3. Rationalization (Conjugate Method)

If the function involves square roots, multiply the numerator and denominator by the conjugate of the radical expression to simplify the limit.

4. L'Hôpital's Rule

If a limit results in 0/0 or ∞/∞, you can take the derivative of the numerator and the derivative of the denominator separately and then evaluate the limit again.

How This Calculator Works

This calculator uses numerical estimation. It evaluates the function at points extremely close to the target (e.g., a – 0.000001 and a + 0.000001). If both sides converge to the same value, it provides that value as the limit. This is helpful for complex functions where algebraic manipulation is difficult.