Skip to content
Step-by-step calculus

Limit Calculator

Enter an expression, the point it approaches and the direction, and get the limit with the reasoning explained.

Works in your browserSteps explainedVideo explanations

Learn the method

A limit describes an approach

The value near a point can exist even when the function is undefined at the point itself.

lim⁡x→1x2−1x−1=lim⁡x→1(x+1)=2\lim_{x\to1}\frac{x^2-1}{x-1}=\lim_{x\to1}(x+1)=2
Calculus formulas

Worked example

  1. 1Direct substitution produces 0/0, which is an indeterminate form.
  2. 2Factor x²−1=(x−1)(x+1). For x≠1, cancel the common factor.
  3. 3Now x+1 approaches 2. The original function still has a hole at x=1.

Matching sampled values suggest a trend; they do not prove that a limit exists. Check both one-sided limits and any domain restrictions.

What this limit calculator does

  • Limits at a finite point or at ±∞, two-sided or one-sided (from the left or the right)
  • Explains direct substitution, factoring, L'Hôpital's rule, squeeze theorem and series as needed
  • Says clearly when a limit does not exist or diverges, and why
  • Graph of the approach with a table of values on both sides of the point

How to use it

  1. Type the expression. Use ∞ or inf for infinity, abs(x) or |x| for absolute value.
  2. Enter the point the variable approaches and choose two-sided, left or right.
  3. Press Solve to see the limit, the steps and the approach on the graph.

Questions about the limit calculator

Can the limit calculator find one-sided limits?

Yes. Choose 'from the left' or 'from the right'. For two-sided limits the solver reports both one-sided limits when they differ.

Does it handle limits at infinity?

Yes. Enter ∞ or -∞ as the point. The solver explains dominant terms, horizontal asymptotes and growth comparisons.

What does 'does not exist' mean?

The one-sided limits differ, the function oscillates, or it grows without bound. The solver states which case applies and shows the one-sided values.

How is the limit verified?

The expression is evaluated at points closer and closer to the limit point from each side, and the trend is compared with the stated limit.