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Limit Calculator

Evaluate limits of functions as x approaches any value or infinity. See numerical approach table, direct substitution, and step-by-step method.

lim
x →
sin(x)/x(x²-1)/(x-1)(1+1/x)^xpolye^xln(x)
Limit
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Limits: The Foundation of Calculus

A limit describes the value a function approaches as its input approaches some value. Limits are the rigorous foundation upon which derivatives, integrals, and continuity are defined. The notation lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x gets close to a. Limits handle cases where direct evaluation fails, such as 0/0 indeterminate forms, and they define what happens at boundaries, discontinuities, and infinity.

Methods for Evaluating Limits

1. Direct Substitution: plug in x=a
  If f(a) is defined, lim = f(a)

2. Factoring: simplify 0/0 forms
  (x²-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1

3. L'Hôpital's Rule (0/0 or ∞/∞):
  lim f/g = lim f'/g'

4. Squeeze Theorem:
  If g(x)≤f(x)≤h(x) and lim g=lim h=L
  then lim f = L

5. Known Limits:
  lim(x→0) sin(x)/x = 1
  lim(x→∞) (1+1/x)^x = e
Indeterminate Forms
Seven indeterminate forms require special techniques: 0/0, ∞/∞, 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞. Direct substitution gives no answer for these. Use algebraic manipulation, L'Hôpital's Rule, or series expansion to resolve them.

Frequently Asked Questions

What is a limit in calculus? ▾
A limit describes the value a function approaches as its input approaches a specific value. It does not require the function to actually equal that value at the point.
What is an indeterminate form? ▾
An expression like 0/0 or ∞/∞ that does not determine the limit by itself. Special techniques like L'Hôpital's Rule or algebraic simplification are needed.
What is L'Hôpital's Rule? ▾
If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists. Take derivatives of top and bottom separately.
Can a limit exist if the function is undefined? ▾
Yes. The limit depends on what f(x) approaches, not its actual value. Example: f(x)=(x²-1)/(x-1) is undefined at x=1, but its limit as x→1 is 2.
What is a one-sided limit? ▾
Approaching from only one direction. lim(x→a⁺) means from the right; lim(x→a⁻) means from the left. A two-sided limit exists only if both one-sided limits are equal.