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.
Limit Calculator
Select a limit problem below to see the step-by-step solution, convergence table, and method explanation.
Select a Limit Problem
Famous Standard Limits
Rational Functions
Trigonometric Limits
Exponential & Logarithmic
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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) = LHow to Evaluate Limits
Try direct substitution first: plug the approach value into the function
If you get a number, that's your limit — you're done
If you get 0/0, try factoring and canceling common terms
For 0/0 or ∞/∞ that can't be factored, apply L'Hôpital's Rule
For limits at infinity, compare polynomial degrees or divide by highest power
Check one-sided limits if the two-sided limit doesn't exist
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.
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.
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.
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.
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.
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.
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.
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.