Limit Calculator

Q: How do I enter infinity in the limit calculator?

To enter infinity in the limit point field, type 'oo' (two letter o's), 'inf', or 'infinity'. For negative infinity, use '-oo', '-inf', or '-infinity'. You can also use 'pi' for π and 'e' for Euler's number.

Calculate limits of mathematical functions with detailed step-by-step solutions. Supports one-sided limits, indeterminate forms, and L'Hospital's Rule.

Limit Calculator

Quick Examples

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Function f(x) - use sin, cos, exp, ln, sqrt, ^

Variable

Limit Point (a) - use oo for ∞, -oo for -∞, pi, e

Direction of Approach

Both Sides Two-sided limit

From Right x → a⁺

From Left x → a⁻

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About Limit Calculator

Welcome to the Limit Calculator, your comprehensive tool for computing mathematical limits with detailed step-by-step solutions. Whether you are a student learning calculus, a teacher preparing lessons, or a professional needing quick limit calculations, this calculator provides accurate results with clear explanations of each step.

What is a Limit in Calculus?

A limit describes the value that a function approaches as the input (usually denoted as $x$) approaches a particular value. The concept of limits is fundamental to calculus and forms the foundation for understanding derivatives, integrals, and continuity.

Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value $L$ that $f(x)$ gets arbitrarily close to as $x$ gets arbitrarily close to $a$. Formally written as: $$\lim_{x \to a} f(x) = L$$

Types of Limits

Two-Sided Limits

A two-sided limit considers the behavior of the function as $x$ approaches $a$ from both the left and right sides. For the limit to exist, the function must approach the same value from both directions:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

One-Sided Limits

Left-hand limit (from the left): $\lim_{x \to a^-} f(x)$ - The value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$
Right-hand limit (from the right): $\lim_{x \to a^+} f(x)$ - The value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$

Limits at Infinity

We can also evaluate limits as $x$ approaches positive or negative infinity to understand the long-term behavior of functions:

$$\lim_{x \to \infty} f(x) \quad \text{or} \quad \lim_{x \to -\infty} f(x)$$

Indeterminate Forms

When direct substitution results in an undefined expression, we encounter an indeterminate form. These require special techniques to evaluate:

Form	Description	Common Solution
0/0	Zero divided by zero	L'Hospital's Rule, Factoring, Rationalization
∞/∞	Infinity divided by infinity	L'Hospital's Rule, Divide by highest power
0 × ∞	Zero times infinity	Rewrite as 0/0 or ∞/∞
∞ - ∞	Infinity minus infinity	Combine fractions, Rationalization
0⁰	Zero to the power of zero	Logarithmic transformation
1^∞	One to the power of infinity	Logarithmic transformation
∞⁰	Infinity to the power of zero	Logarithmic transformation

L'Hospital's Rule

L'Hospital's Rule is a powerful technique for evaluating limits that result in indeterminate forms of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$:

L'Hospital's Rule

If $\lim_{x \to a} \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then: $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$ provided the limit on the right exists. This rule can be applied repeatedly if necessary.

How to Use This Limit Calculator

Enter the function: Type your mathematical function in the expression field. Use standard notation like sin(x), cos(x), e^x, ln(x), x^2, sqrt(x), etc.
Specify the variable: Enter the variable used in your function (usually x). This can be any letter like t, n, or theta.
Enter the limit point: Type the value that the variable approaches. Use "oo" for infinity, "-oo" for negative infinity, or any number like 0, 1, pi.
Choose the direction: Select whether to calculate a two-sided limit (both sides), right-hand limit (from the right), or left-hand limit (from the left).
Calculate and review: Click "Calculate Limit" to see the result. Review the step-by-step solution to understand how the limit was computed.

Common Limits to Know

Here are some fundamental limits that appear frequently in calculus:

$\displaystyle\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ (The sinc limit)
$\displaystyle\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$
$\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
$\displaystyle\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ (Definition of $e$)
$\displaystyle\lim_{x \to 0^+} x \ln(x) = 0$
$\displaystyle\lim_{x \to \infty} \frac{\ln(x)}{x} = 0$ (Logarithms grow slower than polynomials)

Input Syntax Guide

When entering expressions, use the following syntax:

Basic operations: +, -, *, /, ^ (power)
Functions: sin(x), cos(x), tan(x), exp(x) or e^x, ln(x), log(x), sqrt(x)
Constants: pi, e, oo (infinity)
Parentheses: Use parentheses to group expressions: (x^2 - 4)/(x - 2)

Frequently Asked Questions

What is a limit in calculus?

A limit describes the value that a function approaches as the input approaches a particular value. It is denoted as $\lim_{x \to a} f(x)$ and is fundamental to calculus, forming the basis for derivatives and integrals.

What is an indeterminate form?

An indeterminate form occurs when direct substitution in a limit gives an undefined expression like 0/0, ∞/∞, 0×∞, ∞-∞, 0^0, 1^∞, or ∞^0. These forms require special techniques like L'Hospital's Rule or algebraic manipulation to evaluate.

What is L'Hospital's Rule?

L'Hospital's Rule states that for limits of the form 0/0 or ∞/∞, the limit of f(x)/g(x) equals the limit of f'(x)/g'(x), where f' and g' are the derivatives. This rule can be applied repeatedly until the indeterminate form is resolved.

What is the difference between one-sided and two-sided limits?

A two-sided limit considers the function's behavior as x approaches a value from both directions. One-sided limits only consider approach from one direction: left-hand limit (x→a⁻) or right-hand limit (x→a⁺). A two-sided limit exists only if both one-sided limits exist and are equal.

How do I enter infinity in the limit calculator?

To enter infinity in the limit point field, type "oo" (two letter o's), "inf", or "infinity". For negative infinity, use "-oo", "-inf", or "-infinity". You can also use "pi" for π and "e" for Euler's number.