limLimit Calculator

Compute limits at a point or at infinity with full step-by-step solutions.

Enter Expression
limx→0
Variable: Limit Point: Use inf for ∞, pi for π
Direction:
Steps: Full working
Try these examples
$sin(x)/x$ x→0 $(x^2-4)/(x-2)$ x→2 $(1-cos(x))/x$ x→0 $(sqrt(x+1)-1)/x$ x→0 $exp(x)/x^2$ x→inf $(x^3-1)/(x-1)$ x→1 $ln(x)/(x-1)$ x→1 $x*sin(1/x)$ x→inf $(2*x^2+3)/(x^2-1)$ x→inf $tan(x)/x$ x→0 $(x^2-9)/(x-3)$ x→3 $sin(3*x)/x$ x→0
Steps Modes
None  Clean answer only.
Summary  Pill list of methods applied.
Detailed  Accordion: before → after, formula & explanation per step.
Use one-sided directions to detect jump discontinuities.
Limit Techniques
Direct Sub.$f(a) if defined$
Factor/Cancel$0/0 → cancel (x−a)$
Conjugate$√a form → multiply conjugate$
L'Hôpital$0/0 or ∞/∞ → f'/g'$
Degree Analysis$rational, x→∞$
sin(x)/x$→ 1 as x→0$
(1−cosx)/x$→ 0 as x→0$
(1+1/x)^x$→ e as x→∞$
Squeeze Thm.$if f≤g≤h, limf=limh=L$
One-sided$lim⁺ ≠ lim⁻ → DNE$
Input Tips
  • Limit point: 0, inf, -inf, pi
  • sin(x)/x → classic trig limit
  • (x^2-1)/(x-1) → factoring
  • sqrt(x+1)-1)/x → conjugate
  • One-sided: select Left or Right direction
  • Use ^ for powers, * to multiply
Fundamental Limits
\(\lim_{x\to 0}\frac{\sin x}{x}\)= 1
\(\lim_{x\to 0}\frac{1-\cos x}{x}\)= 0
\(\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x\)= e
\(\lim_{x\to 0}(1+x)^{1/x}\)= e
\(\lim_{x\to 0}\frac{e^x-1}{x}\)= 1
\(\lim_{x\to 0}\frac{\ln(1+x)}{x}\)= 1
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