Limit Calculator

The Limit Calculator evaluates limits as x approaches a finite value, infinity, or from one side. It identifies the form first. If direct substitution works, it says so and stops. If the form is indeterminate, 0/0, infinity/infinity, 0 times infinity, 1^infinity, or infinity minus infinity, it chooses a method: factor and cancel, rationalize, use a standard limit like sin(x)/x, apply the squeeze theorem, or apply L'Hopital's rule with the derivatives shown. When one-sided limits disagree it reports that the limit does not exist and shows both sides. Limits of piecewise functions at the break point, limits involving absolute values, and limits of sequences written as functions of n are treated the same way, with the form identified first.

Type the limit in plain words or notation: "lim x->0 of (sin 3x)/x", "limit as x approaches infinity of (3x^2 + 1)/(x^2 - 4)", "lim x->2+ of 1/(x-2)". A photo of the printed limit works. If the course has not covered L'Hopital yet, say "without L'Hopital" and the solver uses algebraic methods only. Set Answer form to exact for symbolic values and decimal when you want a numeric approximation. For a limit defined by a graph or a table of values, describe the values or photograph the table and say which point is being approached. For an epsilon-delta proof, say "prove using the definition" and the solver writes the proof with delta chosen explicitly.

Output: the limit value, or a statement that it does not exist or is infinite, first. Then the form identified at substitution, the method chosen, and each algebraic step to the final value. L'Hopital steps show the derivatives of numerator and denominator separately. The working is shown so you can check it. For the derivative itself, use the Derivative Calculator. For convergence of a sequence or series, use the Series & Sequences Solver. Ask a follow-up to see the same limit computed numerically from a table of nearby values, which is a good check on the algebraic result. Continuity questions at a point use the same three-part test and are answered in the same format.