Limit Calculator

Reviewed by CalcMulti Editorial Team·Last updated: March 2026·← Calculus Hub

A limit describes the value a function approaches as the input approaches a specific number or infinity — even if the function is undefined at that exact point. Limits are the rigorous foundation beneath all of calculus: derivatives are defined as limits of difference quotients, and definite integrals are limits of Riemann sums.

Written lim[x→a] f(x) = L, this means f(x) gets arbitrarily close to L as x approaches a (from either side). For the limit to exist, the left-hand limit lim[x→a⁻] must equal the right-hand limit lim[x→a⁺].

Evaluation techniques include direct substitution, factoring/simplification, rationalization, and L'Hôpital's rule for indeterminate forms (0/0 or ∞/∞). Limits at infinity — where x → ±∞ — reveal horizontal asymptotes of functions.

Worked example: lim[x→2] (x² − 4)/(x − 2). Direct substitution gives 0/0 (indeterminate). Factor: (x² − 4)/(x − 2) = (x+2)(x−2)/(x−2) = x+2 for x ≠ 2. So the limit is 2 + 2 = 4.