Direct substitution
When the function is defined at the point, replace x with it. For 2x + 1 at x = 3 the answer is 2·3 + 1 = 7.
Limit Calculator
Find the limit of a function as x approaches a number or infinity, from both sides or from one side. Type a function, say where x is heading, and see the answer, the steps, a table of values, and a graph.
Result
Checked: the table of values moves toward this limit.
Step-by-step solution
Direct substitution
Replacing x with that value makes both the numerator and the denominator 0. The form 0/0 has no value on its own, so another method is needed.
L'Hôpital's Rule
The form is 0/0, so L'Hôpital's Rule applies: differentiate the numerator and the denominator separately.
Substitute again
Now the limit can be found directly.
Table of values
The value of f(x) at points closer and closer to where x is heading. The calculator uses these numbers to check its answer.
| x | f(x) |
|---|---|
| -0.1 | 2.955202067 |
| -0.01 | 2.99955002 |
| -0.001 | 2.9999955 |
| -0.0001 | 2.999999955 |
| x | f(x) |
|---|---|
| 0.1 | 2.955202067 |
| 0.01 | 2.99955002 |
| 0.001 | 2.9999955 |
| 0.0001 | 2.999999955 |
Graph
The graph appears here once there is a valid function.
What is a limit?
A limit describes where a function is heading as its input gets close to a value. The function does not need to be defined at that value. For example, (x² − 4) / (x − 2) cannot be evaluated at x = 2 because it gives 0/0, yet its values get closer and closer to 4 as x approaches 2, so the limit is 4. Limits are the foundation of calculus: they define continuity, derivatives, and integrals, and they describe asymptotes on a graph. This free limit calculator finds two-sided, left-hand, right-hand, and infinite limits, shows the method used at each step, and checks the answer with a table of values and a graph.
How to use the limit calculator
- Type your function. Use x as the variable, for example
(x^2-4)/(x-2). The keys under the box insert common symbols and functions. - Check the preview. The line "We read this as" shows the limit exactly as it was understood, so you can spot a missing bracket early.
- Enter the value x approaches. Type a number such as
2, an expression such aspi/2, orinfor-inffor infinity. - Choose the direction. Pick both sides for the usual limit, or the left or the right for a one-sided limit.
- Read the result. You get the answer, a step-by-step solution, a table of values on each side, and a graph with the limit point marked.
Types of limits
| Type | Written as | What it means |
|---|---|---|
| Two-sided limit | x → a | The value f(x) approaches as x gets close to a from both sides. |
| Left-hand limit | x → a⁻ | The value f(x) approaches using only x smaller than a. |
| Right-hand limit | x → a⁺ | The value f(x) approaches using only x larger than a. |
| Infinite limit | f(x) → ∞ | The values grow without bound near a point. The graph has a vertical asymptote. |
| Limit at infinity | x → ∞ | The value f(x) approaches as x becomes very large or very negative. The graph may have a horizontal asymptote. |
Methods for finding a limit
Always try direct substitution first. If it gives a number, you are done. If it gives 0/0 or ∞/∞, use one of the methods below. These examples come straight from the calculator.
Factor and cancel
Substituting x = 2 into (x² − 4) / (x − 2) gives 0/0. Factor the top as (x − 2)(x + 2), cancel x − 2, and substitute again: x + 2 is 4 at x = 2.
Rationalize with a conjugate
For (√(x + 1) − 1) / x, multiply the top and bottom by √(x + 1) + 1. This gives 1 / (√(x + 1) + 1), which is 1/2 at x = 0. The calculator reaches the same value with L’Hôpital’s Rule.
L’Hôpital’s Rule
For 0/0 or ∞/∞, differentiate the top and the bottom separately and take the limit again. For sin(3x) / x the new fraction is 3cos(3x) / 1, which is 3 at x = 0.
Compare degrees at infinity
For a ratio of polynomials as x → ∞, look at the highest powers. When the degrees match, the limit is the ratio of the leading coefficients, here 3/5.
An infinite limit
The values of 1/x² grow without bound on both sides of 0, so the limit is ∞. The graph has a vertical asymptote at x = 0.
One-sided limits that differ
|x| / x is −1 just left of 0 and 1 just right of 0. The two sides disagree, so the two-sided limit does not exist.
Common limits
These standard results appear in almost every calculus course. You can enter any of them to see the steps.
When a limit does not exist
- The left-hand and right-hand limits are different, like |x| / x at 0.
- The function swings up and down forever without settling, like sin(1/x) at 0.
- The values grow toward ∞ on one side and −∞ on the other, like 1/x at 0.
- The function is not defined on one side of the point, like ln(x) at 0.
What this calculator can and cannot do
The calculator works with one variable, x, and handles these functions:
- Polynomials, powers, roots, and fractions, such as x³, √x, and 1/(x − 1).
- Exponential and logarithmic functions, such as eˣ, 2ˣ, and ln(x).
- Trigonometric, inverse trigonometric, and hyperbolic functions, and absolute value.
It finds exact answers with direct substitution, factoring and cancelling, comparing degrees, and L’Hôpital’s Rule. When none of these applies, it estimates the limit from nearby values and marks the answer with ≈. It does not do limits of several variables, piecewise functions, or proofs with the squeeze theorem. Fractional powers of negative numbers, such as x^(1/3) to the left of 0, are treated as undefined.
Tips for working with limits
- Substitute first. Many limits are just the value of the function.
- A result of 0/0 means "needs more work", not "does not exist".
- Check both sides when the function has an absolute value or a denominator that changes sign.
- Use brackets for fractions:
(x^2-4)/(x-2)is not the same asx^2-4/x-2. - Look at the table of values. If the numbers settle toward one value, your answer is probably right.
Limit terms explained
- Limit
- The value a function approaches as its input approaches a given number or infinity.
- One-sided limit
- A limit taken from only the left or only the right of a point.
- Indeterminate form
- An expression such as 0/0 or ∞/∞ that cannot be evaluated by substitution alone.
- Vertical asymptote
- A vertical line the graph gets closer to while the values grow without bound.
- Horizontal asymptote
- A horizontal line the graph approaches as x goes to ∞ or −∞.
- Continuous function
- A function whose limit at a point equals its value there, so you can find the limit by substitution.
- L’Hôpital’s Rule
- A method for 0/0 and ∞/∞ forms: differentiate the top and bottom separately and take the limit again.
Related calculators
The derivative is defined as a limit, so the derivative calculatoris a natural next step. To go the other way, try the integral calculator.
Frequently asked questions
What is a limit in calculus?
A limit is the value a function gets closer and closer to as its input gets closer to some number. The function does not have to be defined at that number. For example, (x² − 4) / (x − 2) cannot be evaluated at x = 2, but its values approach 4 as x approaches 2, so the limit is 4.
How do I find a limit?
Start by substituting the value into the function. If you get a number, that is the limit. If you get 0/0 or ∞/∞, try another method: factor and cancel a common factor, multiply by a conjugate to remove a square root, or use L’Hôpital’s Rule. For limits at infinity, compare the highest powers. This calculator uses direct substitution, factoring and cancelling, comparing degrees, and L’Hôpital’s Rule, and it shows the ones it used.
What does 0/0 mean when I find a limit?
It is called an indeterminate form. It does not mean the limit does not exist. It only means that substitution alone cannot tell you the answer, because the top and the bottom are both shrinking to zero and the result depends on how fast each one does. Factoring, conjugates, or L’Hôpital’s Rule can resolve it.
What is a one-sided limit?
A one-sided limit looks at the function from only one direction. The left-hand limit, written x → a⁻, uses values smaller than a. The right-hand limit, written x → a⁺, uses values larger than a. The ordinary two-sided limit exists only when both one-sided limits exist and are equal.
Can a limit be infinite?
Yes. If the values of the function grow without bound near a point, we write that the limit is ∞ (or −∞ if they decrease without bound). This is a way of describing how the function behaves, since ∞ is not a number. On a graph it shows up as a vertical asymptote, as with 1/x² at x = 0.
What are limits at infinity?
A limit at infinity asks what value f(x) approaches as x becomes very large (x → ∞) or very negative (x → −∞). If the answer is a number L, the graph has a horizontal asymptote at y = L. For example, 1/x approaches 0 as x → ∞.
When does a limit not exist?
A two-sided limit does not exist when the left-hand and right-hand limits are different (as with |x|/x at 0), when the function swings up and down without settling (as with sin(1/x) at 0), when it grows toward ∞ on one side and −∞ on the other (as with 1/x at 0), or when the function is not defined on one side of the point.
How do I type infinity or a point such as π/2?
In the "x approaches" box, type inf for infinity or -inf for negative infinity. For other values, type a number such as 2 or -3.5, or an expression such as pi/2 or sqrt(2). In the function, use x as the variable and type it as you would on a calculator, for example sin(3x)/x or (x^2-4)/(x-2).
Does the calculator show the steps?
Yes. It lists each method it used, such as direct substitution, factoring and cancelling, comparing degrees, or L’Hôpital’s Rule, along with the expression at each stage. It also shows a table of values on each side of the point and a graph so you can see that the answer makes sense.
Why does the calculator show an approximate answer?
When none of the exact methods apply, the calculator estimates the limit from values of the function very close to the point and marks the result with ≈. Limits such as (1 + 1/x)ˣ as x → ∞ (which is e) are handled this way. Treat an estimate as a guide and confirm it by hand if you need an exact form.