Limit Calculator (Numerical)
This calculator numerically evaluates limits by approaching the limit point from both sides and checking whether the values converge to the same number. Enter f(x) as a JavaScript expression and the limit point c (or select infinity). The calculator evaluates f at c plus and minus 0.001, 0.0001, and 0.00001 to estimate the left and right limits, then reports whether the limit exists and what value it approaches. For example, to evaluate lim(x to 0) of sin(x)/x, enter Math.sin(x)/x and set c = 0. The calculator shows values approaching 1.000 from both sides, confirming the limit is 1. The limit of f(x) as x approaches c exists when the left limit and right limit both exist and agree. Supported syntax: +, -, *, /, ** (power), Math.sin, Math.cos, Math.tan, Math.exp, Math.log, Math.sqrt, Math.abs, and Math.PI. Use x as the variable. Limits are fundamental to the definition of the derivative and the definite integral in calculus.
Approach table
| x (from left) | f(x) | x (from right) | f(x) |
|---|
Limit evaluation method
Left limit: approach c from x < c (c - 0.001, c - 0.0001, c - 0.00001)
Right limit: approach c from x > c (c + 0.001, c + 0.0001, c + 0.00001)
Limit exists if left limit = right limit
For infinity: evaluate at x = 1e6, 1e9, 1e12
Limits: frequently asked questions
What is a limit in calculus?
The limit of f(x) as x approaches c is the value that f(x) gets closer and closer to as x approaches c (but is not necessarily equal to f(c)). Written lim(x->c) f(x) = L. For the limit to exist, f(x) must approach the same value L from both the left (x approaching c from below) and the right (x approaching c from above).
What is a one-sided limit?
The left-hand limit lim(x->c-) f(x) is the limit as x approaches c from values less than c. The right-hand limit lim(x->c+) f(x) is the limit as x approaches c from values greater than c. The two-sided limit exists if and only if both one-sided limits exist and are equal.
What does 0/0 at a limit point mean?
When substituting c into f gives 0/0 (an indeterminate form), the limit may still exist. For example, lim(x->0) sin(x)/x = 1, even though sin(0)/0 = 0/0. Numerically, this calculator evaluates f close to c to estimate the limit without substitution. Algebraically, L'Hopital's rule handles such cases.
What is the limit as x approaches infinity?
The limit of f(x) as x approaches infinity is the value that f(x) approaches as x grows without bound. This calculator evaluates f at large x values (1e6, 1e9, 1e12) when 'infinity' is selected. For example, lim(x->inf) 1/x = 0, and lim(x->inf) (1+1/x)^x = e.
When does a limit not exist?
A limit does not exist if: the left and right limits differ; f(x) oscillates infinitely (like sin(1/x) as x->0); or f(x) grows without bound (like 1/x as x->0, which approaches +infinity from the right and -infinity from the left). This calculator flags when left and right numerical estimates differ significantly.
Sources
- Limits and continuity: NIST DLMF 1.4, Calculus of One Variable.
- L'Hopital's rule and indeterminate forms: Wolfram MathWorld, L'Hopital's Rule.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.