Limit Calculator

Find the limit of a function as x approaches a number or infinity, from either side, with the method shown: substitution, factorising, L'Hôpital's rule or leading terms.

f(x) =	(x^2 - 1)/(x - 1)
as x approaches	1
From	Both sides

f(x) =

Use x, + − * / ^, brackets, sin, cos, tan, ln, log, sqrt, exp, abs, π and e.

as x approaches

A number, pi, or inf / -inf for infinity.

From

lim x→1 (x^2 − 1) / (x − 1)

As x gets closer to 1, f(x) gets closer to 2 (found by factorise and cancel).

Exact fraction. Checked numerically.

Method: Factorise and cancel
Worked out with: AllCalcHub symbolic rules
Check: Every step compared at several values of x; f evaluated ever closer to the point from each side, and the values settle on the answer

Left-hand limit: 2
Right-hand limit: 2
Value at the point itself: undefined

Values of f(x) approaching the point

Distance h	f(a − h)	f(a + h)
±0.1	1.9	2.1
±0.01	1.99	2.01
±0.001	1.999	2.001
±0.0001	1.9999	2.0001

y = (x^2 − 1) / (x − 1) near the point

Limitat (1, 2)Point at the chart, or Tab to it and use ← →

Limit: (1, 2)

About the Limit Calculator

A limit asks what value f(x) approaches as x gets closer to a point, whether or not f is defined there. If substituting gives a number, that's the limit. If it gives 0/0, rewrite first: factorise and cancel, or use L'Hôpital's rule (differentiate the top and the bottom separately).

At infinity, compare the highest powers. Every answer is also checked by evaluating f closer and closer to the point from each side, the table you see below, so you can watch the values settle.

L'Hôpital's rule is another route for 0/0 and ∞/∞ forms: differentiate the top and the bottom separately, then take the limit again.

The formula

$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$

lim x→0 sin x / x: 1
lim x→0 (eˣ − 1)/x: 1
lim x→∞ (1 + 1/x)ˣ: e

Worked example

lim x→1 (x² − 1)/(x − 1)

Substitution gives 0/0
Factorise: (x − 1)(x + 1)/(x − 1) = x + 1
Substitute: 1 + 1 = 2

Questions people ask

What does 'does not exist' mean?

The values approach different numbers from the left and the right (like |x|/x at 0), grow without bound in opposite directions, or never settle (like sin(1/x) at 0).

Is the limit the same as the value?

Only for continuous functions. (x² − 1)/(x − 1) isn't defined at x = 1, but its limit there is 2.