# What is the limit of a function?

Finds the limit of a function at a number or infinity, checked from both sides.

- Page: https://www.acalculator.org/math/limit-calculator
- JSON spec: https://www.acalculator.org/math/limit-calculator.json
- Version: 84661ce56949

## Default answer

Example with the default inputs (Function f(x) (x^2 - 3x)/(2x^2 - 5x - 3), x approaches 3): The limit of (x^2 - 3x)/(2x^2 - 5x - 3) as the variable approaches 3 is 3/7.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The function, in one letter. |
| a | x approaches | A number, or inf or -inf. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| limit | Limit | The limit, written exactly. |
| value | As a decimal | The limit to 10 significant figures. |

## Method

A computer algebra system finds the limit; it shows only after f is checked near the point on each side.

## Assumptions

- The limit is two-sided and finite.
- One variable letter (x if none); radians; ln is the natural logarithm.
- An answer that fails its check is not shown.

## Worked examples

1. f = (x^2 - 3x)/(2x^2 - 5x - 3), a = 3 gives limit = 3/7, value = 0.428571. Source: OpenStax Calculus Vol. 1, 2.3, Ex. 2.17. https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws.
2. f = 1/x + 5/(x (x - 5)), a = 0 gives limit = -1/5, value = -0.2.
3. f = (3x - 1)/(2x + 5), a = inf gives limit = 3/2, value = 1.5.
4. f = (sqrt(x + 2) - 1)/(x + 1), a = -1 gives limit = 1/2, value = 0.5.

## FAQ

### What is a limit?

The limit of f(x) as x approaches a is the value f(x) gets close to when x gets close to a, from both sides, whatever f does at a itself. (x² − 9)/(x − 3) is not defined at 3, but for x near 3 it equals x + 3, so its limit as x → 3 is 6.

### How do I enter infinity?

Type inf, infinity, ∞ or oo for +∞, and -inf for −∞. The limit of (3x − 1)/(2x + 5) as x → ∞ is 3/2: for large x the leading terms 3x/2x dominate.

### Why do I get no answer for 1/x at 0?

The page gives two-sided limits only. 1/x grows without bound to the right of 0 and falls without bound to the left, so the two-sided limit does not exist. Infinite limits (1/x² at 0 is +∞) are not given either. Where f is defined on one side only, as √x at 0, the limit from that side is given.

### How is the limit checked?

The page evaluates f at a ± 0.0001, 0.00001 and 0.000001 (scaled by the size of a) on each side where f is a real number. At every one of those points f must be within 0.1% of the limit and get closer as x gets closer. For x → ∞ the points are 10,000, 100,000 and 1,000,000. A limit that fails is not shown.

### What is the difference between the limit and the value at the point?

The value f(a) is what f gives at a; the limit is what f approaches near a. For a continuous function such as x² + 1 they are equal. They differ at a hole, a jump, or a point where f is not defined: sin(x)/x has no value at 0, but its limit there is 1.

### Why does the page say "No verified answer"?

The algebra system found no formula, its answer failed the numeric check, or the work took over 3 seconds. Some correct limits still fail the check, because f cannot be computed accurately that close to a: (1 − cos x)/x² near 0 loses its digits to rounding. The page does not guess.

## Sources

- OpenStax, Calculus Volume 1, section 2.3 The Limit Laws: https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws
- OpenStax, Calculus Volume 1, section 4.6 Limits at Infinity and Asymptotes: https://openstax.org/books/calculus-volume-1/pages/4-6-limits-at-infinity-and-asymptotes