Limit Calculator

Calculate the limit of any function as x approaches a value or infinity. Checks one-sided limits, detects indeterminate forms, and verifies with epsilon-delta method.

Limit value (numerical approx.)
—
Left limit (x→c⁻) —
Right limit (x→c⁺) —
Limit exists (left ≈ right)? —
Extended More scenarios, charts & detailed breakdown ▾
Two-sided limit (numerical approx.)
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Indeterminate form? —
Professional Full parameters & maximum detail ▾

Limit Result

Limit (numerical approx.) —
L'Hopital applicable? —

Verification

ε-δ check: |f(c±1e-5) − L| < 1e-4? —
Sequence approach: f(c + 1/n) for n=1000 —

How to Use This Calculator

  1. Enter f(x) (e.g. (x^2-1)/(x-1)).
  2. Enter the value c that x approaches.
  3. The calculator evaluates left and right one-sided limits and checks agreement.
  4. Use the Limit at ±∞ tab for limits at infinity.
  5. The Professional tab adds L'Hôpital detection and ε-δ verification.

Formula

Left limit: lim(x→c⁻) f(x) ≈ f(c − ε)  |  Right limit: lim(x→c⁺) f(x) ≈ f(c + ε)

Limit exists iff left = right.

Example

lim(x→1) (x²−1)/(x−1) = lim(x→1)(x+1) = 2. Both one-sided limits ≈ 2.000000.

Frequently Asked Questions

  • The calculator evaluates f(c − ε) and f(c + ε) for a very small ε (10⁻⁷) and averages the results. If both one-sided values agree, the two-sided limit exists.
  • Forms like 0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞ are indeterminate — the limit may exist but requires special techniques like L'Hôpital's rule or algebraic simplification.
  • If lim f/g = 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the derivative limit exists. The calculator flags when this may apply.
  • Use the 'Limit at ±∞' tab and evaluate f(10¹²) or f(−10¹²) as a proxy for the infinite limit. Rational functions p(x)/q(x) → ratio of leading coefficients.
  • If the left limit (x→c⁻) and right limit (x→c⁺) are different, the two-sided limit does not exist at c. Example: lim(x→0) |x|/x = ±1 from each side.

Related Calculators

📐 Derivative Calculator ∫ Integral Calculator Σ Taylor Series Calculator
Sources & References (5) ▾
  1. Limits — Paul's Online Math Notes — Lamar University
  2. Limits and Continuity — Khan Academy
  3. MIT OCW 18.01 Limits — MIT
  4. Limit — Wolfram MathWorld — Wolfram MathWorld
  5. Calculus Volume 1 (OpenStax) — OpenStax