Tooldeon

Limit calculator

Type f(x) and the point x approaches, or inf for infinity. The tool evaluates f at points closer and closer to it, reports what the values settle on, and shows the table. It is a numeric estimate, not a symbolic proof.

Use x, numbers, + - * / ^, parentheses, pi, e and sin(x), cos(x), tan(x), exp(x), ln(x), log(x) (base 10), sqrt(x), abs(x).
A number or constant such as 0, 2, -1.5 or pi/2. Type inf or -inf for infinity.
Limit (estimate)1
From the left
1
From the right
1
f at the point itself
not defined
from the left, x = -0.1
f = 0.998334166
from the left, x = -0.01
f = 0.999983333
from the left, x = -0.001
f = 0.999999833
from the left, x = -0.0001
f = 0.999999998
from the left, x = -0.00001
f = 1
from the left, x = -0.000001
f = 1
from the left, x = -1e-7
f = 1
from the left, x = -1e-8
f = 1
from the right, x = 0.1
f = 0.998334166
from the right, x = 0.01
f = 0.999983333
from the right, x = 0.001
f = 0.999999833
from the right, x = 0.0001
f = 0.999999998
from the right, x = 0.00001
f = 1
from the right, x = 0.000001
f = 1
from the right, x = 1e-7
f = 1
from the right, x = 1e-8
f = 1

This is a numeric estimate from 8 sample points per side, at distances 0.1 down to 0.00000001 (or 10 up to 100,000,000). A limit can look settled and still be different, so treat the result as a strong hint and confirm it by algebra when it matters.

Runs in your browser: nothing you enter is sent anywhere.

How to use it

  1. Type f(x).
  2. Type the point, or inf / -inf.
  3. Choose both sides, or one side to get a one-sided limit.
  4. Read the estimate and look at the table to see the values closing in.

How it works

For each side the tool builds eight x values moving toward the point and evaluates f at each. If the last four values grow quickly past 100,000 the result is reported as infinite. Otherwise the stretch where successive values differ least is taken as the limit when it is stable to about five digits, and values within one millionth of a whole number are rounded to it. When both sides are computed they must agree.

Worked example

sin(x)/x as x → 0. At x = ±0.1 the value is 0.998334, at ±0.01 it is 0.999983, and by ±0.0001 it is 0.99999999833. The estimate is 1.

Tips

  • Use the one-sided options for jumps, absolute values and functions with a vertical asymptote.
  • For a limit at infinity try both inf and -inf separately.
  • To confirm exactly, simplify algebraically or use the Taylor series calculator.

Frequently asked questions

How is the limit estimated?

f is evaluated at a ± 0.1, 0.01, … 0.00000001 (or at 10, 100, … 100,000,000 for infinity). The tool picks where neighbouring values change least, which avoids both the far points and the last ones damaged by rounding, and rounds the result.

Why can the answer be wrong?

Numeric work cannot prove anything. Functions that oscillate, such as sin(1/x) near 0, or that change only at a scale smaller than the samples, can fool it. Rounding errors also limit how close x can get to the point.

What does it show for 1/x at 0?

From the right the values grow without bound, +∞; from the left they fall to −∞. The sides disagree, so the two-sided limit does not exist.

Does the value at the point matter?

No. A limit looks at nearby points, not at the point itself, which is why sin(x)/x has limit 1 at 0 even though f(0) is not defined. The tool shows f at the point only for comparison.

More in Math

All tools in this category: Math ·Updated November 3, 2026

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