Limit Calculator
Limit of f(x) as x approaches a number, +∞ or −∞: f(c) where f is continuous, otherwise an estimate from samples. Shows one-sided limits, ±∞ and jumps. How it works
Use the Limit Calculator
About the Limit Calculator
The limit of sin(x)/x as x approaches 0 is 1, and the limit of (x² − 1)/(x − 1) as x approaches 1 is 2. This limit calculator gives f(c) where f is continuous and otherwise estimates the limit of f(x) as x approaches a number, +∞ or −∞ from samples on each side, shows the left-hand and right-hand values, and says when the limit is infinite, does not exist or cannot be decided.
Limits as x approaches a number
| f(x) as typed | x → | From the left | From the right | Tool shows | Exact limit |
|---|---|---|---|---|---|
| x^2 | 2 | 4 | 4 | 4 | 4 |
| sin(x) | 1 | 0.84147 | 0.84147 | 0.84147 | sin 1 ≈ 0.841471 |
| sin(x)/x | 0 | 1 | 1 | 1 | 1 |
| (x^2-1)/(x-1) | 1 | 2 | 2 | 2 | 2 |
| (x^3-8)/(x-2) | 2 | 12 | 12 | 12 | 12 |
| (1-cos(x))/x^2 | 0 | 0.5 | 0.5 | 0.5 | 1/2 |
| (exp(x)-1)/x | 0 | 1 | 1 | 1 | 1 |
| (sqrt(x+4)-2)/x | 0 | 0.25 | 0.25 | 0.25 | 1/4 |
| (1+x)^(1/x) | 0 | 2.7183 | 2.7183 | 2.7183 | e ≈ 2.71828 |
| x*sin(1/x) | 0 | 0 | 0 | 0 | 0 |
| 1/x^2 | 0 | +∞ | +∞ | +∞ (no real-number limit) | +∞ |
| 1/x | 0 | −∞ | +∞ | does not exist | Does not exist |
| abs(x)/x | 0 | -1 | 1 | does not exist | Does not exist |
| floor(x) | 1 | 0 | 1 | does not exist | Does not exist |
| sqrt(x) | 0 | not defined (no real values) | 0 | does not exist | Does not exist (right-hand limit 0) |
| sin(1/x) | 0 | does not settle on a value | does not settle on a value | does not exist | Does not exist |
In the first two rows f is continuous, so the tool shows f(c) itself. The others are estimates, printed with the digits their error estimate supports (at most 5 significant). Type the first column into the Function box as it is written.
Limits as x goes to infinity
| f(x) as typed | x → | Tool shows | Exact limit |
|---|---|---|---|
| (1+1/x)^x | +∞ | 2.7183 | e ≈ 2.71828 |
| (3x^2+1)/(x^2-5) | +∞ | 3 | 3 |
| (2x+1)/(x-3) | −∞ | 2 | 2 |
| sqrt(x^2+x)-x | +∞ | 0.5 | 1/2 |
| sin(x)/x | +∞ | 0 | 0 |
| exp(x) | −∞ | 0 | 0 |
| x^2 | +∞ | +∞ (no real-number limit) | +∞ |
| sin(x) | +∞ | does not exist | Does not exist |
| exp(x) | +∞ | +∞ (no real-number limit) | +∞ |
| log(x) | +∞ | +∞ (no real-number limit) | +∞ |
| sqrt(x)-sqrt(x-1) | +∞ | 0 | 0 |
Where the tool is wrong or cannot decide
| f(x) as typed | x → | Tool shows | Correct answer |
|---|---|---|---|
| x+1e-10/x | 0 | 0 | Does not exist (−∞ from the left, +∞ from the right) |
| 1+1e-9*sign(x) | 0 | 1 | Does not exist (a jump of 0.000000002) |
| sin(x)/(x+1e-13) | 0 | 1 | 0 |
| (1-cos(x))/x^2 | 1e-7 | 0.4996 | About 0.5 |
| floor(x) | 1e12 | 1000000000000 | Does not exist (a jump) |
| 1/(1+1e-15x) | +∞ | 1 | 0 |
| exp(x)/x^200 | +∞ | 0 | +∞ |
| (1-1/x)^(x^2) | +∞ | 1 | 0 |
| gamma(x) | −∞ | 0 | Does not exist (a pole at every negative integer) |
| (1-cos(x))/x^2 | 1e-8 | cannot decide | About 0.5 |
| x^2 | 1e308 | cannot decide | 10^616, too large to store |
| sin(x) | 1e13 | cannot decide | sin(10^13) |
| log(log(log(x))) | +∞ | cannot decide | +∞ |
| x*sin(x) | +∞ | cannot decide | Does not exist (oscillates) |
| x^100/exp(x) | +∞ | cannot decide | 0 |
| log(x)^10/x | +∞ | cannot decide | 0 |
| sin(x^2) | +∞ | cannot decide | Does not exist (oscillates) |
| round(x)-x | +∞ | cannot decide | Does not exist (oscillates) |
The wrong answers come from features much smaller than about 10⁻⁸ of the sampled values, from cancellation between nearly equal numbers, and from overflow.
Methodology
The tool computes f(c) first: if f is continuous at c, the limit is f(c) itself. Otherwise it samples f on each side at 37 steps that halve from 0.2371 × |c| (0.2371 at c = 0) down to about 3 × 10⁻¹² of it (for infinity, out to 10³⁰⁰), extrapolates to a zero step with Ridders’ and Aitken’s methods and prints only the digits the error estimate supports, at most 5 significant. Two sides must agree within their error estimates for a limit to be shown.
How to use this tool
- Type f(x) with x as the variable, for example (x^2-1)/(x-1). 2x is read as 2*x, sin, cos and tan take radians, and log is the natural log (log10 and log2 are separate functions).
- Under What x approaches, pick a number c and type it, or pick +∞ or −∞. Or press one of the six examples, such as sin(x)/x at 0, to fill in all three.
- Read the limit and the accuracy line under it. For a number c the left-hand and right-hand values are always shown as well. The graph under the result shows f from c − 1 to c + 1, with open circles at the values each side approaches.
Frequently asked questions
- How do you find a limit step by step?
- Try substituting first. If that gives 0/0, simplify: (x² − 1)/(x − 1) = x + 1 for x ≠ 1, so the limit at 1 is 2. The tool does not show algebra steps. It finds that f has no value at 1, samples f on both sides at steps that halve from about 0.24 down to about 3 × 10⁻¹², and shows 2 from both sides, which checks your working.
- What is the limit of sin(x)/x as x approaches 0?
- 1, from both sides. Two related limits come out the same way: tan(x)/x at 0 is 1 and (1 − cos(x))/x² at 0 is 0.5.
- How do you know when a limit does not exist?
- Compare the two sides. abs(x)/x at 0 gives -1 from the left and 1 from the right, a jump. 1/x at 0 goes to −∞ on the left and +∞ on the right. sin(1/x) at 0 keeps oscillating and settles on nothing. 1/x² goes to +∞ from both sides: the tool shows +∞, which means there is no limit as a real number.
- How do you find a limit at infinity?
- Choose x approaches +∞ or −∞. The tool samples x from about 2.4 out to 10³⁰⁰, so fast growth such as exp(x) or log(x) shows as +∞. For a ratio of polynomials of the same degree the limit is the ratio of the leading coefficients: (3x² + 1)/(x² − 5) goes to 3. (1 + 1/x)^x goes to e, shown as 2.7183.
- Can it find one-sided limits?
- Yes. For a number c it shows the limit from the left and from the right separately. sqrt(x) at 0 is not defined on the left and 0 on the right, so only the right-hand limit exists. floor(x) at 1 is 0 from the left and 1 from the right.
- Why does it say it cannot decide?
- Because the samples on at least one side are too noisy, too few or too slow to settle, and the tool will not guess. x^2 at 1e308 gives this answer, because x² is too large to store near that point. Try a simpler form of the function, for example with common factors cancelled, or check by hand.
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