By The limit calculator finds \( \lim_{x \to c} f(x) \) for any function you type, whether \( c \) is a number or \( \pm\infty \), and shows how it got there. It is built for calculus students checking homework. It is free and needs no sign-up.
Every answer comes with its working: direct substitution when the function is continuous, L’Hôpital’s rule when you hit \( \tfrac{0}{0} \) or \( \tfrac{\infty}{\infty} \), and a table of values when neither symbolic method applies.
How to use the limit calculator
- Type the function of \( x \). Use
^for powers andsqrt(),sin(),cos(),ln(),abs(),e^(x)andpifor the usual functions. Implicit multiplication works, so3xandx sin(x)are fine. - Enter the value that \( x \) approaches. Type
infor-inffor limits at infinity. - For a finite point, choose the direction: two-sided, from the left or from the right.
- Press Solve step by step. The result appears first, followed by the numbered steps. Use Copy result to grab the answer or Plot in Visual Lab to see the graph near the point.
How it works
The calculator tries three methods, in this order.
1. Direct substitution. If \( f \) is continuous at \( c \), then \( \lim_{x\to c} f(x) = f(c) \). The tool checks that \( f(c) \) is defined and that values just to the left and right of \( c \) agree with it.
2. L’Hôpital’s rule. If \( f = \frac{g}{h} \) and substitution gives \( \tfrac00 \) or \( \tfrac{\infty}{\infty} \), then
$$\lim_{x\to c}\frac{g(x)}{h(x)} = \lim_{x\to c}\frac{g'(x)}{h'(x)}$$
as long as the right side exists. The calculator differentiates the top and bottom symbolically, repeats while the form is still indeterminate (up to six rounds), then substitutes. This step is used at finite points.
3. Numerical approach. When neither method applies, for example a limit at infinity or a product like \( x\ln x \), the tool evaluates \( f \) at points closing in on \( c \) (\( c \pm 0.1, 0.01, 0.001, \dots \)) or at growing values of \( |x| \) up to \( 10^8 \). It shows the first rows of that table and reports the value the outputs settle on. If the left and right sides settle on different values, the answer is “does not exist”. Treat this as a reliable estimate rather than a proof.
Worked example
Example 1: a \( \tfrac00 \) form. Find \( \lim_{x\to2}\frac{x^3-8}{x-2} \).
Substituting gives \( \frac{8-8}{2-2} = \frac00 \), which tells you nothing yet. Differentiate the top and the bottom separately: \( (x^3-8)' = 3x^2 \) and \( (x-2)' = 1 \). So
$$\lim_{x\to2}\frac{x^3-8}{x-2} = \lim_{x\to2}\frac{3x^2}{1} = 3(2)^2 = 12.$$
Factoring confirms it: \( x^3 - 8 = (x-2)(x^2+2x+4) \), so for \( x \ne 2 \) the fraction is \( x^2+2x+4 \), which is 12 at \( x = 2 \). Type (x^3 - 8)/(x - 2) with \( x \to 2 \) into the calculator above to see the L’Hôpital step.
Example 2: a limit at infinity. Find \( \lim_{x\to\infty}\frac{6x^2 - x}{3x^2 + 4} \).
Divide the top and bottom by \( x^2 \), the highest power in the denominator:
$$\frac{6 - \frac1x}{3 + \frac{4}{x^2}} \;\to\; \frac{6 - 0}{3 + 0} = 2.$$
Enter (6x^2 - x)/(3x^2 + 4) and inf. The table shows \( f(10) = \frac{590}{304} \approx 1.9408 \) and \( f(100) = \frac{59900}{30004} \approx 1.9964 \), and the values keep closing in on 2, which is the answer reported.
Reading the result when there is no finite limit
A number isn’t the only possible outcome:
- \( \infty \) or \( -\infty \) means the function grows without bound, with the same sign on both sides. For example, \( \lim_{x\to0}\frac{1}{x^2} = \infty \).
- Does not exist means the one-sided limits disagree, or one side never settles. For \( \frac{x^2-1}{|x-1|} \) at \( x = 1 \), the left-hand limit is \( -2 \) and the right-hand limit is \( 2 \). The calculator lists both values so you can see why, and the one-sided limit calculator lets you evaluate each side on its own.
- Domain edges need a direction. \( \sqrt{x} \) is not defined for \( x < 0 \), so a two-sided limit at 0 comes back as “does not exist”. Choose “from the right” and you get 0. The same goes for \( x\ln x \) at 0, whose right-hand limit is 0.
Trig functions use radians. For a deeper look at the rule in step 2, read the guide to L’Hôpital’s rule or try the L’Hôpital’s rule calculator. For end behavior and asymptotes, see limits at infinity.
Related guides
- L’Hôpital’s Rule: When and How to Use It
- Limits at Infinity: Rules, Asymptotes and Examples
- Limit of sin(x)/x as x → 0 Equals 1: Three Proofs
- Squeeze Theorem: Statement and Examples
Further reading
- Computing Limits (Paul’s Online Math Notes) — factoring, rationalizing and the Squeeze Theorem for limits substitution can’t handle.
- Limits at Infinity, Part I (Paul’s Online Math Notes) — end behavior of polynomials and rational functions, and horizontal asymptotes.
FAQ
Can the limit calculator show steps?
Yes. Every result lists the method used: direct substitution, each round of L’Hôpital’s rule with the new numerator and denominator, or a table of function values approaching the point.
How do I enter a limit at infinity?
Type inf (or -inf) in the “x approaches” box. Limits at infinity are found numerically by evaluating \( f \) at increasingly large \( x \), so the direction selector isn’t needed.
Why does the calculator say the limit does not exist?
Usually the left-hand and right-hand limits differ, the function oscillates, or it isn’t defined on one side of the point. Switch the direction to “from the left” or “from the right” to see each one-sided limit on its own.
Is the limit calculator free?
Yes. It runs in your browser, has no usage limits and needs no account.