What is L'Hôpital's Rule?

L'Hôpital's Rule states that if lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the right side exists. You differentiate the numerator and denominator separately (not using the quotient rule) and take the limit again. You may need to apply it multiple times.

What are the seven indeterminate forms?

The seven indeterminate forms are: 0/0, ∞/∞, 0×∞, ∞-∞, 0^0, 1^∞, and ∞^0. These forms don't have an automatic answer — the actual limit depends on how fast each part approaches its value. Special techniques are needed to resolve them.

Why does lim(x→0) sin(x)/x = 1?

This is a famous standard limit proved geometrically using the squeeze theorem. As x approaches 0, sin(x) behaves almost exactly like x (they have the same Taylor series leading term). Using L'Hôpital's Rule: d/dx(sin x) = cos x, d/dx(x) = 1, so the limit is cos(0)/1 = 1.

When does a limit not exist?

A limit doesn't exist when: (1) the left-hand and right-hand limits are different, (2) the function oscillates infinitely as it approaches the point, (3) the function goes to +∞ from one side and -∞ from the other. The limit can also be ±∞, which means it exists as an infinite limit.

What is the limit definition of e?

Euler's number e ≈ 2.71828 is defined as lim(x→∞) (1 + 1/x)^x. This limit involves the indeterminate form 1^∞. It appears naturally in compound interest (continuous compounding) and is the base of the natural logarithm.

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Limit Calculator

Calculate limits step-by-step with direct substitution, factoring, and L'Hôpital's rule. Includes famous limits like sin(x)/x and (1+1/x)^x.

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Calculate Limits Step-by-Step

Evaluate limits of functions with our free limit calculator. Solve indeterminate forms like 0/0, ∞/∞, and 0×∞ using factoring, L'Hôpital's Rule, and standard limit identities. Perfect for calculus students preparing for AP Calculus, college exams, or learning limit evaluation techniques.

What is a Limit?

A limit describes the value that a function approaches as the input approaches some value. Written as lim(x→a) f(x) = L, it means that f(x) gets arbitrarily close to L as x gets close to a. Limits are fundamental to calculus, defining derivatives, integrals, and continuity. When direct substitution yields an indeterminate form like 0/0 or ∞/∞, special techniques are needed to evaluate the limit.

Limit Notation

lim(x→a) f(x) = L

Why Use This Calculator

Free Step-by-Step Solutions

See exactly how each limit is solved with detailed explanations of the method used.

Learn Multiple Techniques

Master factoring, L'Hôpital's Rule, and standard limit identities through examples.

Convergence Tables

Visualize how function values approach the limit from left and right.

Famous Limits Library

Study essential limits like sin(x)/x = 1 and (1+1/x)^x = e with full explanations.

Identify Indeterminate Forms

Recognize 0/0, ∞/∞, 0×∞, 1^∞, and other forms that require special handling.

Perfect for Exam Prep

Ideal for AP Calculus, College Calculus I/II, and standardized math tests.

How to Evaluate Limits

Applications of Limits

Defining Derivatives

The derivative f'(x) is defined as the limit of (f(x+h) - f(x))/h as h approaches 0.

Defining Integrals

The definite integral is defined as a limit of Riemann sums.

Continuity Analysis

A function is continuous at a point if the limit equals the function value.

Asymptote Identification

Horizontal and vertical asymptotes are found using limits at infinity and undefined points.

Frequently Asked Questions

Start with direct substitution. If that gives a number, you're done. If it gives an indeterminate form like 0/0, try factoring to cancel common terms, or use L'Hôpital's Rule (differentiate top and bottom separately). For limits at infinity, compare the degrees of polynomials or divide everything by the highest power of x.

What is a Limit?

Limit Notation

lim(x→a) f(x) = L