Limit Calculator
The tool below evaluates a limit — the value a function approaches as its input gets close to a particular point (or grows without bound). Enter a limit in standard notation and the calculator picks the right technique: direct substitution, factoring, L'Hôpital's rule, or one of the common special limits.
Type naturally or click ⌨ in the box to use the math keyboard.
Use \lim_{x \to a} for the limit notation. For infinity, type \infty.
Examples: \lim_{x \to 2} (x^2 + 3x), \lim_{x \to 0} \frac{\sin x}{x}.
A limit describes the value a function approaches as its input gets close to a target. For example, \(\lim_{x \to 3}(x^2 + 3x) = 18\) — as \(x\) gets close to 3, the expression gets close to 18. Limits underpin both derivatives and integrals, and they're how calculus handles values that can't be evaluated directly (like dividing by zero).
Worked Examples
Three examples worked by hand — the same logic the calculator uses, just spelled out so you can follow it on paper.
Example 1: Direct substitution — evaluate \(\lim_{x \to 2}(x^2 + 3x)\)
When the function is continuous at the target point, just substitute. The expression \(x^2 + 3x\) is a polynomial, so it's continuous everywhere:
\[\lim_{x \to 2}(x^2 + 3x) = (2)^2 + 3(2) = 4 + 6 = 10\]
Direct substitution is always the first thing to try. It only fails when substitution gives an indeterminate form like \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\).
Example 2: Factoring through an indeterminate form — evaluate \(\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}\)
Substituting \(x = 3\) gives \(\tfrac{0}{0}\), which is indeterminate. Factor the numerator instead:
\[\dfrac{x^2 - 9}{x - 3} = \dfrac{(x-3)(x+3)}{x-3} = x + 3 \quad \text{(for } x \neq 3\text{)}\]
Now substitution works on the simplified form:
\[\lim_{x \to 3}(x + 3) = 3 + 3 = 6\]
The cancellation is legal under the limit because the limit only cares about values near \(x = 3\), not at \(x = 3\) itself.
Example 3: L'Hôpital's rule — evaluate \(\lim_{x \to 0} \dfrac{\sin x}{x}\)
Substituting \(x = 0\) gives \(\tfrac{0}{0}\). When direct substitution fails and the form is \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), L'Hôpital's rule says the limit equals the limit of the derivatives:
\[\lim_{x \to 0} \dfrac{\sin x}{x} = \lim_{x \to 0} \dfrac{\cos x}{1} = \dfrac{\cos(0)}{1} = 1\]
This particular limit (\(\sin x / x \to 1\) as \(x \to 0\)) shows up so often it's worth memorizing.
What is a Limit?
A limit captures the value a function approaches as its input gets arbitrarily close to a target. The notation \(\lim_{x \to a} f(x) = L\) means: as \(x\) gets close to \(a\) (from either side), \(f(x)\) gets close to \(L\).
Limits handle situations where direct evaluation breaks down. The expression \(\tfrac{x^2 - 9}{x - 3}\) is undefined at \(x = 3\) (division by zero), but the limit as \(x \to 3\) still exists and equals 6 — the value the function would have if you "patched" the hole.
Limits are the foundation of calculus: the derivative is a limit of average rates of change, and the definite integral is a limit of Riemann sums.
Try These Examples
\lim_{x \to 2}(x^2 + 3x)— direct substitution\lim_{x \to 3} \frac{x^2 - 9}{x - 3}— indeterminate, solved by factoring\lim_{x \to 0} \frac{\sin x}{x}— classic L'Hôpital\lim_{x \to \infty} \frac{3x^2 + x}{x^2 - 1}— limit at infinity\lim_{x \to 0^+} \ln x— one-sided limit going to negative infinity\lim_{x \to 0}(1 + x)^{1/x}— definition of \\(e\\)
Common Techniques for Evaluating Limits
A quick reference for the methods this calculator uses:
| Technique | When to use |
|---|---|
| Direct substitution | First thing to try. Works whenever the function is continuous at the target. |
| Factoring | Substitution gives \(\tfrac{0}{0}\) and the numerator (or denominator) factors. |
| Rationalizing | The expression involves a square root and substitution gives \(\tfrac{0}{0}\). |
| L'Hôpital's rule | Form is \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\). Take the derivative of top and bottom, then try again. |
| Compare degrees | Limit at infinity of a rational function. The ratio of leading terms tells you the answer. |
| Special limits | \(\lim_{x \to 0} \tfrac{\sin x}{x} = 1\), \(\lim_{x \to 0}(1+x)^{1/x} = e\), and a few others worth memorizing. |
Tips for Using the Calculator
- For the limit notation, type
\lim_{x \to a}wherexis the variable andais the target value - For infinity, use
\infty(or pick it from the virtual keyboard) - For one-sided limits, use
x \to a^+(right side) orx \to a^-(left side) - For powers, use
^:x^2means \(x^2\) - For fractions, use
\frac{a}{b}or justa/b - The virtual math keyboard has a
limtemplate for one-click entry
If your problem is about finding where a function is defined rather than its limiting behavior, see the lesson on finding limits for more context.
Frequently Asked Questions
What's an indeterminate form?
An indeterminate form is an expression like \(\tfrac{0}{0}\), \(\tfrac{\infty}{\infty}\), \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), or \(1^\infty\) that doesn't have a fixed value — the answer depends on how the parts approach their limits. These are signals to switch from direct substitution to a technique like factoring, rationalizing, or L'Hôpital's rule.
When do I use L'Hôpital's rule?
L'Hôpital's rule applies when direct substitution gives the indeterminate form \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\). In that case, the limit of the ratio equals the limit of the ratio of derivatives: \(\lim \tfrac{f(x)}{g(x)} = \lim \tfrac{f'(x)}{g'(x)}\). You can apply it more than once if the new ratio is still indeterminate.
What's a one-sided limit?
A one-sided limit only considers values approaching the target from one direction. \(\lim_{x \to a^+} f(x)\) considers values greater than \(a\) (the right side); \(\lim_{x \to a^-} f(x)\) considers values less than \(a\) (the left side). A two-sided limit exists only when both one-sided limits exist and are equal.
What if the limit doesn't exist?
A limit doesn't exist when the left and right one-sided limits disagree, when the function oscillates indefinitely near the point (like \(\sin(1/x)\) at \(x = 0\)), or when the function grows without bound. The calculator will say "the limit does not exist" or indicate \(\pm\infty\) as appropriate.
What's the difference between a limit and an asymptote?
A horizontal asymptote is the value a function approaches as \(x \to \pm\infty\). A vertical asymptote at \(x = a\) means the function grows without bound as \(x\) approaches \(a\) (the one-sided limit is \(\pm\infty\)). Asymptotes are a visual description of the graph; limits are the algebraic mechanism that defines them.