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
You might also find these calculators useful
Solve quadratic equations and find roots
Calculate logarithms: natural (ln), common (log10), binary, and custom base
Calculate powers and exponentials: a^b, e^x, 10^x, 2^x, and nth roots
Calculate percentages, percentage change, and more
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.
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) = LSee exactly how each limit is solved with detailed explanations of the method used.
Master factoring, L'Hôpital's Rule, and standard limit identities through examples.
Visualize how function values approach the limit from left and right.
Study essential limits like sin(x)/x = 1 and (1+1/x)^x = e with full explanations.
Recognize 0/0, ∞/∞, 0×∞, 1^∞, and other forms that require special handling.
Ideal for AP Calculus, College Calculus I/II, and standardized math tests.
The derivative f'(x) is defined as the limit of (f(x+h) - f(x))/h as h approaches 0.
The definite integral is defined as a limit of Riemann sums.
A function is continuous at a point if the limit equals the function value.
Horizontal and vertical asymptotes are found using limits at infinity and undefined points.
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.