Calculator
Use z as the complex variable and i for the imaginary unit. Supported functions: sin cos tan exp log sqrt abs arg conj re im.
Formula used
A complex limit lim(z→a) f(z) = L means: for every ε>0 there exists δ>0 such that 0<|z−a|<δ ⇒ |f(z)−L|<ε.
This tool estimates the limit numerically by sampling f(z) as z approaches a along several paths. If the last values on each path stabilize (path tolerance) and the path estimates agree (agreement tolerance), the limit is reported as likely existing.
- Radii: rₖ = r₀·shrinkᵏ
- Line approach: z = a + r·e^{iθ}
- Power: z^w = exp(w·log(z)) (principal branch)
How to use this calculator
- Enter f(z) using z and i.
- Set the approach point a with real and imaginary parts.
- Pick paths, steps, and shrink factor to reach smaller radii.
- Adjust tolerances if results look unstable or too strict.
- Compute, review path agreement, then export CSV or PDF.
FAQs
1) Does this prove the limit exists?
No. It provides numerical evidence by testing many approaches and checking agreement. For a proof, use ε–δ reasoning, algebraic simplification, or analytic theorems.
2) What expressions are supported?
Use z, constants i pi e, operators + - * / ^, parentheses, and functions sin cos tan exp log sqrt abs arg conj re im.
3) Why do different paths matter?
A complex limit must be the same along every approach to the point. If two paths give different stable values, the limit does not exist at that point.
4) What do “path tolerance” and “agreement tolerance” mean?
Path tolerance checks whether the last sampled values on a path stabilize. Agreement tolerance checks whether stabilized path estimates match each other closely enough.
5) How should I choose r₀, steps, and shrink?
Start with r₀ around 0.1–1, steps 8–12, shrink 0.5–0.8. If values blow up, reduce r₀. If convergence is slow, add steps or use a smaller shrink.
6) Why can log or power behave oddly?
Complex log uses a principal branch, so it has discontinuities across a branch cut. That can affect z^w and may show path-sensitive behavior.
7) What if I get “division by zero”?
Your function may be undefined near the approach point. Try a smaller r₀, fewer extreme paths, or simplify the expression analytically to remove removable singularities.
8) How do the downloads work?
After a computation, the latest inputs, path estimates, and sample values are saved in your session. The CSV and PDF buttons export that stored result instantly.