How to use it
- Type the function, for example sin(x)/x.
- Enter the point x approaches (a number, pi/2, inf or -inf).
- Choose two-sided, left-hand or right-hand.
- Click Find limit and read the result and the table.
How the limit is found
The calculator works two ways at once:
- Symbolically, using a computer algebra engine, to find an exact value such as 1, 1/2 or e.
- Numerically, by evaluating the function closer and closer to the point from the left and from the right and watching where the values settle.
If both agree, you get the exact value, confirmed. If the left and right values differ, the two-sided limit doesn't exist, and the calculator says so with both one-sided values. If the values grow without bound, the limit is +∞ or −∞. If they oscillate or the function isn't defined there, it reports that the limit probably doesn't exist.
Classic examples
| Limit | Value | Why |
|---|---|---|
| sin(x)/x as x → 0 | 1 | A fundamental trig limit |
| (1 − cos x)/x² as x → 0 | 1/2 | Taylor series of cos x |
| (x² − 1)/(x − 1) as x → 1 | 2 | Factor and cancel (x − 1) |
| (1 + 1/x)ˣ as x → ∞ | e | Definition of e |
| |x|/x as x → 0 | does not exist | Left: −1, right: +1 |
| x·ln x as x → 0⁺ | 0 | One-sided; ln x undefined for x ≤ 0 |
Click any of these in the examples row to try them.
One-sided limits and infinity
Choose Left-hand (x → a⁻) or Right-hand (x → a⁺) to approach from one side only. If the function is defined on just one side of the point, like ln x or √x at 0, the calculator automatically gives that one-sided limit and tells you.
For limits at infinity, type inf or -inf as the point. The table then shows values for larger and larger |x|.
The table of values
Below the result you see the function evaluated at points approaching the limit from each side, such as 0.1, 0.01, 0.001 and so on. It's a good way to see why the limit is what it is, and it's how limits are introduced in most calculus courses. At very small distances, floating-point rounding can distort values; the estimate uses the most stable ones.
How to enter functions
Type functions the way you'd write them: x^3, sqrt(x) or √x, 2x or 3*x, sin(x)^2 or sin^2(x), e^x or exp(x), ln(x) for the natural log, log10(x), abs(x), and the constants pi and e. Implicit multiplication like 2x, x(x+1) and sinx is understood. The point can be any expression, such as 0, 2, pi/2 or -inf.
Frequently asked questions
What is the limit of sin(x)/x as x approaches 0?
- The calculator confirms it symbolically and with a table of values.
How do I enter infinity?
Type inf or -inf as the point x approaches.
What does it mean when a limit does not exist?
The left- and right-hand limits differ, the function oscillates, or it isn't defined near the point. The calculator tells you which.
Can it do one-sided limits?
Yes. Choose left-hand (x → a⁻) or right-hand (x → a⁺).
Does it use L'Hôpital's rule?
It uses a symbolic engine and a numerical check rather than showing L'Hôpital steps, so it handles 0/0 and ∞/∞ forms but doesn't list the rule applications.
Last updated: November 12, 2026