Limit Calculator
Evaluate limits of functions numerically. Compute left-hand, right-hand, and two-sided limits as x approaches any value. Free online limit calculator for calculus.
About This Calculator
This Limit Calculator evaluates the limit of a function f(x) as x approaches any value a. It computes left-hand limits (x -> a⁻), right-hand limits (x -> a⁺), and the two-sided limit simultaneously using numerical approximation at progressively smaller distances from the target point.
The calculator probes the function at distances of 1, 0.1, 0.01, ... down to 1e-10 from the target value, from both sides separately. It detects convergence (to a finite number), divergence (to +/-∞), or non-existence (oscillation or conflicting one-sided limits).
India (IN): Limits are introduced in CBSE Class 11 mathematics and form the foundation of calculus. JEE Advanced includes limits as a core topic, with problems involving L'Hôpital's rule, rationalization, and standard limit formulas.
United States (US): Limits are taught in AP Calculus AB as the foundation of derivatives and integrals. Students learn to evaluate limits analytically (factoring, conjugates, trig limits) and graphically, including one-sided limits and limits at infinity.
United Kingdom (UK): A-Level Mathematics covers limits in Pure Mathematics as an introduction to calculus. Students learn about the limit of a function, continuity, and the formal definition of the derivative as a limit.
Frequently Asked Questions
What is a limit in calculus?
A limit describes the value that a function f(x) approaches as x gets arbitrarily close to some value a. The notation lim(x->a) f(x) = L means that f(x) can be made as close to L as desired by taking x sufficiently close to a (but not necessarily equal to a).
How does the calculator compute limits numerically?
The calculator evaluates f(x) at progressively smaller distances from the target point a (e.g., a +/- 1, a +/- 0.1, a +/- 0.01, down to a +/- 1e-10). It checks whether the values converge to a specific number, diverge to +/-∞, or fail to settle (does not exist). Separate left and right approaches are computed.
What is the difference between left-hand and right-hand limits?
The left-hand limit lim(x->a⁻) f(x) considers values of x approaching a from below (x < a). The right-hand limit lim(x->a⁺) f(x) considers x approaching a from above (x > a). The two-sided limit exists only when both one-sided limits exist and are equal. For example, f(x) = |x|/x has different left and right limits at x = 0.
What does it mean when a limit does not exist?
A limit does not exist (DNE) when: (1) the left and right limits are different, (2) the function oscillates without settling (like sin(1/x) at x = 0), (3) the function grows without bound in opposite directions, or (4) the function is not defined near the point. The calculator reports DNE in these cases.
What mathematical expressions are supported?
Supported operations: +, -, *, /, ^ (exponentiation), sin(), cos(), tan(), asin(), acos(), atan(), sqrt(), abs(), log()/ln(), exp(), and constants pi and e. Enter expressions like '(x^2 - 1)/(x - 1)' or 'sin(x)/x'.
Why is the limit of (x^2-1)/(x-1) at x=1 equal to 2?
The function (x^2-1)/(x-1) simplifies to (x-1)(x+1)/(x-1) = x+1 for all x ≠ 1. While the function is undefined at x=1 (division by zero), as x approaches 1, the expression approaches x+1 = 2. This classic example shows how limits handle removable discontinuities.