Free Limit Calculator (+ Guide)

What is a limit calculator?

A limit calculator works out the value a function approaches as its input approaches a point, even where the function itself isn't defined.

This one checks each side, toward a number or toward infinity, and shows every value it used.

How it works

The calculator evaluates f at x = a − 0.1, a − 0.01, a − 0.001 and closer, then the same from the right.

Where the values stop changing, that's the limit. Near the point, rounding in the computer's arithmetic starts to show, so the calculator stops before it and extrapolates from the steps it trusts.

Limits from a table of values

OpenStax's Calculus Volume 1 teaches limits the same way, with a table of values. For sin(x)/x near 0, its table reads 0.998334166468 at x = ±0.1, 0.999983333417 at ±0.01 and 0.999999833333 at ±0.001.

The values close in on 1 from both sides, so the limit of sin(x)/x as x approaches 0 is 1, even though sin(0)/0 isn't defined.

One-sided limits

A left-hand limit uses only x below the point, and a right-hand limit only x above it.

OpenStax's Theorem 2.2 ties them together: the limit as x approaches a is L if and only if both one-sided limits are L.

So |x|/x has no limit at 0. From the left it's −1, from the right it's 1, and the calculator shows both.

When a limit doesn't exist

There are three ways a limit can fail to exist, and the calculator says which one it found.

• The sides differ: the left and right limits are different numbers, as with |x|/x at 0
• The values never settle: sin(1/x) swings between −1 and 1 ever faster near 0, as OpenStax shows
• It's infinite: 1/(x − 2)² grows without bound near 2, an infinite limit with a vertical asymptote at x = 2

An infinite limit is written as ∞ or −∞, and when the two sides head in opposite directions, as 1/x does at 0, there's no two-sided limit.

Limits at infinity

As x → ∞, the calculator reads f at 10, 100, 1,000 and further out, the way OpenStax's section on limits at infinity describes horizontal asymptotes.

(3x² + 1)/(x² − 5) heads to 3, and (1 + 1/x)^x heads to e, about 2.7182818.

Worked examples

Here's the example the calculator loads: (x² − 1)/(x − 1) as x approaches 1.

At x = 1 it's 0/0, undefined. But at 0.999 it's 1.999 and at 1.001 it's 2.001, so the limit is 2, the same as x + 1 at 1 once the common factor cancels.

The quick examples above the inputs load four more: sin(x)/x at 0, (1 + 1/x)^x at ∞, 1/x at 0 and (1 − cos x)/x² at 0, which is 1/2.

Mistakes to avoid

These are the slips that come up most.

• Plugging in the point and stopping at 0/0, which says nothing about the limit
• Checking only one side, when the sides can differ
• Using degrees for sin or cos, when calculus limits such as sin(x)/x → 1 assume radians

Who is the limit calculator for?

It's for students in calculus and anyone checking their work on a limit, and these tools help around it.

• Scientific calculator – for evaluating f at a point
• Fraction calculator – for the exact fractions
• Logarithm calculator – for ln and log

Read the table before the answer, and the limit makes sense on its own.