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This free limit calculator finds limits step by step. It tries direct substitution first. If that gives a form like \(\frac{0}{0}\), it uses factoring, L’Hôpital’s rule, or the conjugate. It also handles one-sided limits and limits at infinity, and shows a table of values so you can see the limit with your own eyes.
How to Use the Limit Calculator
- Type the function, for example
sin(x)/x. - Enter the value x approaches. Use
inffor \(\infty\) or-inffor \(-\infty\). - Choose the direction: both sides, from the left (−), or from the right (+).
- Press Solve. Check the steps, the graph, and the table of values.
What Is a Limit?
A limit asks: as x gets closer and closer to a number, what value does \(f(x)\) get close to? We write
\[\lim_{x\to a} f(x) = L.\]
The function does not even need to be defined at \(a\). For example, \(\frac{x^2-1}{x-1}\) is undefined at \(x = 1\), but its values get close to 2. So the limit is 2.
A two-sided limit exists only if the left-hand limit and the right-hand limit are equal.
Methods the Calculator Uses
| Method | When to use it |
|---|---|
| Direct substitution | The function is continuous at the point. Just plug in. |
| Factor and cancel | A fraction of polynomials gives \(\frac00\). |
| L’Hôpital’s rule | You get \(\frac00\) or \(\frac{\infty}{\infty}\). Take derivatives of top and bottom. |
| Compare degrees | A fraction of polynomials as \(x \to \infty\). |
| Multiply by the conjugate | Square roots with \(\infty – \infty\). |
| Use logarithms | Forms like \(1^\infty\), \(0^0\), \(\infty^0\). |
Worked Examples
Example 1: \(\lim_{x\to 1}\frac{x^2-1}{x-1}\)
Plugging in gives \(\frac00\). Factor the top: \(x^2 – 1 = (x-1)(x+1)\). Cancel \((x-1)\) to get \(x + 1\). Now plug in: \(1 + 1 = 2\).
Example 2: \(\lim_{x\to 0}\frac{\sin x}{x}\)
Plugging in gives \(\frac00\). By L’Hôpital’s rule, take derivatives: \(\lim_{x\to0}\frac{\cos x}{1} = 1\).
Example 3: \(\lim_{x\to\infty}\frac{3x^2+1}{2x^2-5}\)
The top and bottom both have degree 2. The limit is the ratio of the leading numbers: \(\frac{3}{2}\).
Example 4: \(\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x\)
This is the \(1^\infty\) form. Let \(y = \left(1+\frac1x\right)^x\), so \(\ln y = x\ln\left(1+\frac1x\right)\). That limit is 1, so \(y \to e^1 = e \approx 2.718\).
Example 5: \(\lim_{x\to 0}\frac{1}{x}\)
From the right, \(\frac1x \to +\infty\). From the left, \(\frac1x \to -\infty\). The sides do not agree, so the two-sided limit does not exist.
Indeterminate Forms
These forms do not tell you the answer on their own. You need more work:
\[\frac{0}{0},\quad \frac{\infty}{\infty},\quad 0\cdot\infty,\quad \infty-\infty,\quad 1^\infty,\quad 0^0,\quad \infty^0\]
A form like \(\frac{5}{0}\) is not indeterminate. It means the function blows up, and you check the sign on each side.
Common Mistakes
- Saying \(\frac00 = 0\) or 1. It is not a number. It means “do more work.”
- Using L’Hôpital’s rule when the form is not \(\frac00\) or \(\frac\infty\infty\). Check first.
- Using the quotient rule in L’Hôpital. Differentiate the top and bottom separately.
- Forgetting one-sided limits. Near a vertical asymptote, the left and right can differ.
Learn the Math Behind the Limit Calculator
Limits describe the value a function approaches. They define the derivative and the definite integral, so they come first in any calculus course.
Frequently Asked Questions
What does the table of values show?
It lists x-values that get closer to the point, and the function’s value at each one. You can watch the numbers settle on the limit.
Can a limit be infinity?
Yes. If \(f(x)\) grows without bound, we write the limit as \(\infty\). Strictly speaking, the limit does not exist as a number, but \(\infty\) tells you how it behaves.
How are limits connected to derivatives and integrals?
Both are defined with limits. The derivative is a limit of slopes. The definite integral is a limit of sums. Improper integrals use limits too. Try the improper integral calculator.