Limit Calculator

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 $\frac{\infty}{\infty}$:

How to Use This Limit Calculator

1. 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.
2. Specify the variable: Enter the variable used in your function (usually x). This can be any letter like t, n, or theta.
3. 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.
4. 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).
5. 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.

References

• Limit (Mathematics) - Wikipedia
• Limits and Continuity - Khan Academy
• L'Hospital's Rule - Wikipedia