Quick answer
A limit describes the value a function approaches as x approaches a point. lim x→0 of sin(x)/x = 1.
Enter a function of x and the value x approaches. The calculator returns the limit and a numeric value.
Related tools: Derivative Calculator: Derivative of a function of x, with a numeric check. Integral Calculator: Antiderivative of a function of x, with optional definite bounds. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Limit Calculator
- Type function of x, for example x^3*sin(x).
- Type x approaches, for example 0, 1, inf, -inf.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Function of x | Text, such as x^3*sin(x) | — |
| x approaches | Text, such as 0, 1, inf, -inf | — |
Formula and method
Formula
lim x→a f(x)
Symbolic result; numeric values to six decimals.
Nerdamer finds the limit symbolically; the value is then confirmed by evaluating the function close to the point (or at very large |x|).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Function of x = sin(x)/x; x approaches = 0.
| Result | Value |
|---|---|
| Limit | 1 |
| Numeric value | 1 |
Example 2
Inputs: Function of x = (x^2-1)/(x-1); x approaches = 1.
| Result | Value |
|---|---|
| Limit | 2 |
| Numeric value | 2 |
Example 3
Inputs: Function of x = (e^x-1)/x; x approaches = 0.
| Result | Value |
|---|---|
| Limit | 1.0 |
| Numeric value | 1 |
The first line is the headline result (limit). Below it the limit Calculator lists numeric value, which describe the same case from other angles. Symbolic result; numeric values to six decimals.
A classic limit
At x = 0 the expression sin(x)/x is 0/0, but values like sin(0.01)/0.01 = 0.99998 approach 1. The limit is proved with the squeeze theorem or l'Hôpital's rule.
Scope
It handles elementary functions of one variable x; anything the engine cannot evaluate is rejected. Limits at infinity, one-sided limits and indeterminate forms need care. Always check with a numeric value.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Function of x: left empty shows "Enter function of x." beside that field and no result.
- x approaches: entering "abc" shows "Enter a number, inf or -inf." beside that field and no result.
Choosing between the Limit Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Derivative Calculator](/math/derivative-calculator/) | function of x and order | Derivative |
| [Integral Calculator](/math/integral-calculator/) | function of x | Antiderivative |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
Background: calculus, vectors and matrices
A derivative measures how fast a function's output changes with its input, and an integral accumulates a quantity such as area. The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc, so [[3, 8], [4, 6]] has determinant 3×6 − 8×4 = −14.
The dot product of two vectors is the sum of the products of matching components and is a single number; a positive value means the vectors point in broadly the same direction.
Assumptions and limitations
- Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported; one-sided limits are not offered
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for function of x and x approaches to confirm this.
Frequently asked questions
What is lim x→∞ 1/x?
As x grows without bound 1/x approaches 0, so the limit is 0.
What is l'Hôpital's rule?
A method for 0/0 and ∞/∞ forms.
Can a limit exist where the function is undefined?
Yes. For sin(x)/x the function is undefined at 0, yet the limit there is 1.
What does the Limit Calculator calculate?
Two-sided limit of a function of x, with a numeric check. It returns Limit and Numeric value.
What formula does the Limit Calculator use?
lim x→a f(x). Nerdamer finds the limit symbolically; the value is then confirmed by evaluating the function close to the point (or at very large |x|).
What do I need to enter in the limit Calculator?
You enter Function of x and x approaches.
How is the result rounded or limited?
Symbolic result; numeric values to six decimals. Handles elementary functions of one variable x; anything the engine cannot simplify or confirm is reported as unsupported; one-sided limits are not offered.
What is an example of a limit result?
With function of x sin(x)/x and x approaches 0, the limit is 1.
Where does the method come from?
The method follows: Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy.
Related calculators
- Derivative Calculator — Derivative of a function of x, with a numeric check.
- Integral Calculator — Antiderivative of a function of x, with optional definite bounds.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Area Calculator — Area of common two-dimensional shapes.
Sources
- Nerdamer symbolic engine, cross-checked numerically against the input; reference values from SymPy
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.