Limit Calculator

Use infinity for ∞ and -infinity for −∞. Both sin3x and sin(3*x) work; oo is also accepted as ∞.
Preview type a function above…
Examples by method
Direct sub
L'Hôpital
Factoring
At infinity
One-sided
Functions: sin, cos, tan, ln (or log), sqrt, e^x, abs(x).
Special points: infinity (or oo) for ∞; -infinity for −∞; pi, e as constants.
One-sided: use the Left/Right toggle above; default is two-sided.
Operators: * for multiplication, / for division, ^ for powers.
lim

Ready when you are

Type a function, set the point it approaches, and hit Calculate Limit.

Calculate a limit to see its graph.

Template:
Direct substitution limx→a f(x) = f(a)  (if continuous)
L'Hôpital lim f(x)/g(x) = lim f′(x)/g′(x)
Squeeze if g ≤ f ≤ h and lim g = lim h = L ⇒ lim f = L

Frequently asked

A limit describes the value a function f(x) approaches as x gets closer to a specific point. Written as limx→a f(x) = L, limits are the foundation of calculus — used to define derivatives, integrals, and continuity. For example, limx→0 sin(x)/x = 1, even though sin(0)/0 is undefined.
(1) Try direct substitution — plug in the value. (2) If you get an indeterminate form like 0/0, factor and cancel common factors. (3) If factoring fails, apply L'Hôpital's Rule by differentiating numerator and denominator separately. (4) For limits at infinity, divide by the highest power of x. This calculator automates all four.
L'Hôpital's Rule states that for indeterminate forms 0/0 or ∞/∞: lim f(x)/g(x) = lim f′(x)/g′(x). Use it when direct substitution gives 0/0 or ∞/∞. Differentiate numerator and denominator separately (NOT via the quotient rule), then re-evaluate. Apply repeatedly if the result is still indeterminate.
The seven indeterminate forms are 0/0, ∞/∞, 0·∞, ∞−∞, 00, 1∞, and ∞0. For 0/0 and ∞/∞ use L'Hôpital or factoring. For 0·∞ rewrite as a fraction. For exponential forms (00, 1∞, ∞0) take the natural log first, then apply L'Hôpital.
A one-sided limit evaluates a function as x approaches a value from only one direction. The left-hand limit (x→a−) uses values slightly less than a; the right-hand limit (x→a+) uses values slightly greater. A two-sided limit exists only if both one-sided limits agree. Use the Left / Two-sided / Right toggle in this calculator to choose.
For rational functions: divide every term by the highest power of x in the denominator. If degrees are equal, the limit is the ratio of leading coefficients. If numerator degree is less, the limit is 0. If greater, the limit is ±∞. For exponentials like ex, the limit as x → ∞ is ∞. Type infinity or -infinity in the Approaches field.
Yes — over 2,000 limit practice problems with full answer keys. Filter by 11 question types (standard limits, one-sided, infinity, L'Hôpital, squeeze, continuity, …) and 4 difficulty levels (basic, medium, hard, scholar). Each worksheet is randomly generated — perfect for exam prep, self-study, or classroom quizzes.
11 question types: standard limits (direct sub, factoring), one-sided with asymptotes, limits at infinity, L'Hôpital (basic and advanced), difference-quotient limits, exponential indeterminate forms (00, 1∞), continuity problems, DNE with absolute value, and squeeze theorem. Range: textbook-style up to scholar-level exam questions.