How it works
The calculator evaluates f(x) at a sequence of points approaching a: x = a ± 0.1, a ± 0.01, a ± 0.001, … down to ±10⁻¹⁰. If those values converge, the limit is reported. If they diverge to ±∞, oscillate, or settle at different values from each side, the calculator reports that explicitly instead.
For limits at infinity (a = ∞ or −∞), the same idea applies but with x = ±10, ±10², ±10³, … up to ±10¹².
Examples to try
| f(x) | a | Limit |
|---|---|---|
sin(x)/x | 0 | 1 |
(1 - cos(x)) / x^2 | 0 | 0.5 |
(1 + 1/x)^x | inf | e ≈ 2.71828 |
1/x | 0 | does not exist (left = −∞, right = +∞) |
sin(1/x) | 0 | oscillates — no limit |
FAQ
Why are some limits reported as "approximate"?
Floating-point arithmetic can't represent x → a exactly. The calculator stops when consecutive values agree to ~10 digits, which is the practical limit of double-precision arithmetic.
Why does my limit say "does not exist" when I think it does?
Most often: left and right limits differ. Look at the trace to see the values from each side. For oscillating functions like sin(1/x), no convergence will ever happen — and that's the correct answer.