Limit Calculator

Calculate limits of functions with steps. Supports one-sided limits and undefined forms.

Our Limit Calculator evaluates the behavior of a function as input approaches a specific value. It handles standard limits, indeterminate forms (0/0, ∞/∞), and limits at infinity.

Perfect for calculus students learning continuity, asymptotic behavior, and L'Hôpital's Rule.

Enter Function & Limit Point

Function f(x)

Approaching (x → ?)

x →

Examples:

Graph Visualization

Calculated Limit

\lim_{x \to 1} \frac{x^2-1}{x-1} = 2

What is Limit Calculator?

A **Limit** describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.

\lim_{x \to a} f(x) = L

This means that as $x$ gets closer and closer to $a$ , $f(x)$ gets closer and closer to $L$ .

Formula & Calculation

Key Limit Rules

Sum Rule $\lim (f + g) = \lim f + \lim g$
Product Rule $\lim (f \cdot g) = \lim f \cdot \lim g$
Quotient Rule $\lim (f / g) = \lim f / \lim g$
L'Hôpital's RuleFor 0/0 or ∞/∞, take derivative of top and bottom.

Example Calculation

Classic Limit Example

Find the limit of $\frac{\sin x}{x}$ as x approaches 0.

Direct substitution gives 0/0. Using geometric proof or L'Hôpital's Rule:

\lim_{x \to 0} \frac{\sin x}{x} = 1

Related Tools

Frequently Asked Questions

What if the limit is 0/0?

This is an "indeterminate form". You cannot say the limit is 0, 1, or undefined yet. You must simplify the expression (factor, conjugate) or use L'Hôpital's Rule.

What is a one-sided limit?

Sometimes a function behaves differently depending on whether you approach from the left (values smaller than a) or the right (values larger than a). If left and right limits don't match, the limit does not exist.

Does the function need to be defined at the point?

No! Limits tell us about the behavior *near* the point, not *at* the point. A function can have a hole at x=a but still have a limit there.