Limit Calculator

Mathematical limits describe the value that a function approaches as its input gets arbitrarily close to a given point — without necessarily reaching that point. Limits are the foundation of calculus: derivatives and integrals are both defined using limits.

Formal definition (epsilon-delta): lim(x→a) f(x) = L means: For every ε > 0, there exists a δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε.

In plain terms: you can make f(x) as close to L as desired by making x sufficiently close to a.

L’Hôpital’s Rule (for indeterminate forms 0/0 or ∞/∞): lim(x→a) [f(x)/g(x)] = lim(x→a) [f’(x)/g’(x)]

Where f’(x) and g’(x) are the derivatives of the numerator and denominator.

Common indeterminate forms:

• 0/0: use L’Hôpital’s or algebraic simplification (factoring, conjugate multiplication)
• ∞/∞: use L’Hôpital’s
• 0 × ∞: rewrite as 0/(1/∞) = 0/0, then apply L’Hôpital’s
• 1^∞, 0^0, ∞^0: use logarithms to reduce to 0/0 form

Standard limits to memorize:

• lim(x→0) sin(x)/x = 1
• lim(x→0) (1 − cos x)/x = 0
• lim(x→0) (e^x − 1)/x = 1
• lim(x→∞) (1 + 1/x)^x = e ≈ 2.71828
• lim(x→∞) (ln x)/x = 0

Worked example using L’Hôpital’s Rule: Evaluate lim(x→0) (sin 3x) / (5x)

Direct substitution: sin(0) / (5 × 0) = 0/0 — indeterminate. Apply L’Hôpital’s:

f(x) = sin 3x → f’(x) = 3 cos 3x g(x) = 5x → g’(x) = 5

lim(x→0) [3 cos 3x / 5] = 3 cos(0) / 5 = 3 × 1 / 5 = 3/5 = 0.6

One-sided limits: lim(x→a⁺) — approaching from the right lim(x→a⁻) — approaching from the left A limit exists only if both one-sided limits are equal. If they differ, the limit does not exist at that point.