Limit Evaluator

Evaluate one-sided and two-sided limits by approaching the point from the left and the right. Detects infinite limits, jumps and oscillation, and recognises exact values such as ½, π and e.

CalculusLimitsContinuity

Inputs

Input examples

f(x) — e.g. sin(x)/x, (x^2−1)/(x−1)

x approaches c — a number, pi, inf or −inf

Direction

What is a Limit?

The limit lim_x→c f(x) = L says that f(x) gets arbitrarily close to L as x gets close to c — without x ever having to equal c. The function does not even need to be defined at c itself.

Approaching from the left (x → c⁻) and from the right (x → c⁺) can give different answers. The two-sided limit only exists when both one-sided limits exist and are equal. A function is continuous at c when the limit equals f(c).

L⁻ = L⁺limit exists

L⁻ ≠ L⁺jump — no limit

f → ±∞vertical asymptote

0/0, ∞/∞indeterminate form

Live Graph

x → 0

Enter a function to see its graph

f(x)

x = c

limit point (c, L)

Result

Enter a function and point above

Standard Limits

Limit	Value

Solving Methods

Direct Substitution

Always try this first. Polynomials, sin, cos, e^x and their sums and products are continuous wherever they are defined.

Example:

Factor & Cancel

When substitution gives 0/0, a common factor usually hides in numerator and denominator. Cancel it, then substitute again.

Example:

L’Hôpital’s Rule

Valid for the indeterminate forms 0/0 and ∞/∞. Differentiate numerator and denominator separately — not as a quotient.

Example:

Sources & Verification

References

J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage, 2016 — Ch. 2 (Limits and Derivatives) and Sec. 4.4 (Indeterminate Forms and l’Hospital’s Rule).
M. Spivak, Calculus, 4th ed., Publish or Perish, 2008 — Ch. 5 (Limits).