Limit Calculator — Evaluate Limits with Step-by-Step

The Formula

The limit of a function describes the value that f(x) approaches as x gets arbitrarily close to a given point a. The formal ε-δ definition states that for any desired tolerance ε > 0, there exists a distance δ > 0 such that whenever x is within δ of a (but not equal to a), f(x) is within ε of L. A limit exists if and only if both the left-hand limit (x → a⁻) and right-hand limit (x → a⁺) exist and are equal.

Variable Definitions

How to Use This Calculator

1. 1

   Enter a function f(x) using x as the variable. Use ^ for powers, sin(), cos(), tan(), sqrt(), pi for π, and e for Euler's number (≈ 2.71828).

2. 2

   Enter the approach value a — the number that x gets close to. This can be any real number.

3. 3

   Select the direction: "Both sides" checks agreement from left and right; "Left" only approaches from below; "Right" only approaches from above.

4. 4

   The calculator evaluates f(x) at points increasingly close to a (steps: 0.1, 0.01, ..., 10⁻⁸), showing the approach sequence in the results.

5. 5

   For "Both sides", the result also reports whether the limit exists (left ≈ right) and thus whether the function is continuous at a.

Quick Reference

Common Applications

• Analyzing function behavior near points where the function is undefined (removable singularities)
• Determining continuity and differentiability of functions in calculus coursework
• Computing derivatives using the limit definition: f'(x) = lim_(h→0) (f(x+h) − f(x)) / h
• Evaluating asymptotic behavior of functions in physics and engineering applications
• Understanding convergence of sequences and series in advanced mathematics

The limit of f(x) as x approaches a exists if and only if the left-hand and right-hand limits both exist and are equal.

Understanding the Concept

The concept of a limit is the foundation of all calculus. Intuitively, the limit asks: what value does f(x) get close to as x gets close to a? The function need not be defined at x = a — in fact, limits are most interesting precisely when the function is undefined at that point (e.g., sin(x)/x at x = 0). This calculator uses a numerical approach: it evaluates the function at a sequence of x-values that approach a from both sides, with incrementally smaller step sizes (0.1, 0.01, 0.001, ..., down to 10⁻⁸). By observing the trend as the step size shrinks, the calculator estimates the limit. For "removable" discontinuities (where the function has a hole but the limit exists), the left and right approaches converge to the same value. For "jump" discontinuities, they converge to different values. For "infinite" discontinuities (vertical asymptotes), the function values diverge to ±∞. A function is continuous at a if f(a) equals the limit — meaning no hole, jump, or asymptote. The formal ε-δ definition provides the rigorous mathematical foundation: for any ε > 0 (no matter how small), there must exist a δ > 0 such that whenever x is within δ of a (but not equal to a), f(x) is within ε of L. This definition captures the idea of "arbitrarily close" with mathematical precision and underlies all of limit theory, including the crucial limit laws: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on — provided each individual limit exists.

Frequently Asked Questions

Sources & References