$Limit Calculator$

Limit Calculator

Find the limit of a function at a point and at infinity: indeterminate forms, L'Hôpital's rule, notable limits and one-sided limits.

Which limit:

Two-sided, x → a

From the left, x → a − 0

From the right, x → a + 0

f(x) =

a =

The point a is a number, an expression (pi, 1/2) or infinity: inf, −inf, ∞. Function examples: sin(x)/x, (x^2 − 1)/(x − 1), (1 + 1/x)^x.

Test yourself

The limit calculator finds the limit of a function at a point and at infinity, as well as one-sided limits. For rational functions it resolves indeterminate forms exactly — by canceling the fraction or dividing by the highest power. For other functions it applies L'Hôpital's rule, and if that does not help it finds the limit numerically and recognizes the answer (for example, e or π/4).

Notable limits

First notable limit	lim_x→0 sin x / x = 1
Second notable limit	lim_x→∞ (1 + 1/x)ˣ = e
Consequences	lim_x→0 (eˣ − 1)/x = 1, lim_x→0 ln(1 + x)/x = 1, lim_x→0 (1 − cos x)/x² = 1/2

Resolving indeterminate forms

0/0 for a ratio of polynomials — factor and cancel: (x² − 4)/(x − 2) = x + 2 → 4 as x → 2.
∞/∞ as x → ∞ — divide by the highest power: the limit is 0, the ratio of the leading coefficients, or ∞ depending on the degrees.
L'Hôpital's rule: for 0/0 or ∞/∞, lim f/g = lim f′/g′.
∞ − ∞ with roots — multiply by the conjugate expression.

Example. lim_x→0 sin x / x: 0/0, by L'Hôpital = lim cos x / 1 = 1.

A limit at a point exists only if the left and right limits agree. For 1/x as x → 0 the left limit is −∞ and the right one is +∞ — there is no two-sided limit.