limLimit Calculator
e.g. sin(x)/x, (x^2-1)/(x-1)
Enter a number (e.g. 0, 1, infinity not supported)
A limit describes the value a function approaches as its input approaches a particular value. Limits are the rigorous foundation of calculus — they underpin the definitions of continuity, derivatives, and integrals. Our calculator uses numerical evaluation from both sides (left and right) to estimate limits for continuous functions.
- 1lim(x→c) f(x) = L means f(x) gets arbitrarily close to L as x approaches c
- 2Left-hand limit (x→c⁻): approach from values less than c
- 3Right-hand limit (x→c⁺): approach from values greater than c
- 4A limit exists only if left and right limits agree
- 5Numerically: evaluate at x = c ± 10⁻⁸ and check if results agree to within 10⁻⁶
lim(x→0) sin(x)/x=1Classic limit; indeterminate 0/0 — use L'Hôpital's rule
lim(x→1) (x²−1)/(x−1)=2Factor: (x+1)(x−1)/(x−1) = x+1 → 2 as x→1
lim(x→0) |x|/x=Does not existLeft limit = −1, right limit = +1
| Law | Formula |
|---|---|
| Sum rule | lim[f+g] = lim f + lim g |
| Product rule | lim[f·g] = lim f · lim g |
| Quotient rule | lim[f/g] = lim f / lim g (if lim g ≠ 0) |
| Power rule | lim[f^n] = (lim f)^n |
| Squeeze theorem | g≤f≤h and lim g=lim h=L ⟹ lim f=L |
| L'Hôpital (0/0 or ∞/∞) | lim f/g = lim f'/g' |
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