Limit Calculator

Find the limit of any function as x approaches a value or infinity — with the left- and right-hand values, a table of approaching values, and a graph. Indeterminate 0/0 forms like sin(x)/x resolve correctly.

Limit —

How to Find a Limit

Enter a function of x and the value x approaches to see the limit and the working behind it.

1

Write your function and the target

Type f(x) using * or juxtaposition for multiplication (2x means 2·x), and function names like sin, cos, ln, exp, and sqrt. Then enter what x approaches — a number like 0, or inf / -inf for infinity.

2

The calculator approaches from both sides

It first tries direct substitution, then samples f(x) at values closing in on the target from the left and the right — never exactly at the point — so 0/0 forms like sin(x)/x settle to their true limit.

3

Read the limit, the sides, and the table

The limit appears at the top, with the one-sided values and a table of f(x) approaching the target. If the two sides disagree, the limit does not exist. Next step in calculus? The Derivative and Integral calculators pick up from here.

Frequently Asked Questions

How do you use this limit calculator?
To use the limit calculator, type your function of x into the f(x) box — for example sin(x)/x, (x²-1)/(x-1), or (1+1/x)ˣ — then enter the value x approaches in the second box. Use a number like 0 or 3, or type inf (or ∞) for a limit at infinity and -inf for negative infinity. The calculator shows the limit, the left- and right-hand values, and a table of f(x) as x approaches the target.
Does it handle 0/0 and other indeterminate forms?
Indeterminate forms such as 0/0 are handled. Because the calculator approaches the target from both sides and never evaluates exactly at the point, forms like sin(x)/x approaching 1, (1 − cos x)/x approaching 0, and (x² − 9)/(x − 3) approaching 6 all resolve correctly. The table of approaching values shows the function settling toward the limit rather than the undefined value at the point itself. It is a numerical method, so a limit that only reveals itself extremely close to the point is reported approximately.
Can it find one-sided limits and limits at infinity?
One-sided limits and limits at infinity are both supported. For a finite target the calculator reports the left-hand limit, with x approaching from below, and the right-hand limit, from above; if the two differ, the two-sided limit does not exist. For infinity, enter inf or -inf and it evaluates f(x) at larger and larger values to find what it settles toward, such as (2x + 1)/(x − 3) approaching 2. Rational functions behave predictably here, since the limit is the ratio of the leading coefficients when the degrees match.
When does a limit not exist?
A limit fails to exist when the left- and right-hand values disagree, as with 1/x at x = 0, which runs to negative infinity from below and positive infinity from above. It also fails when the function grows without bound in both directions, so 1/x² at 0 is reported as an infinite limit, a description of the behaviour rather than a finite value. A third case is oscillation, such as sin(1/x) near 0, which keeps swinging between −1 and 1 without settling. The calculator shows the approaching values so you can see which of these is happening.

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