Calculator Inputs
Enter complex values as a+bi, a-bi, bi, i, -i, or a real number.
Example Data Table
| Case | Function | Approach point | Expected limit | Note |
|---|---|---|---|---|
| Polynomial | z² + 2z + 1 | 1 + i | 3 + 4i | Entire functions accept direct substitution. |
| Removable rational | (z² - 1) / (z - 1) | 1 | 2 | A shared zero cancels analytically. |
| Exponential | 2ez | iπ | -2 | Complex exponentials remain analytic everywhere. |
Formula Used
For analytic families, the calculator applies direct substitution:
Limit rule: limz→z₀ f(z) = f(z₀), whenever f is analytic at z₀.
For rational templates, it also checks the order of zeros in the numerator and denominator. If both vanish to the same order m, the removable form is resolved with matching derivatives:
Removable singularity rule: L = N(m)(z₀) / D(m)(z₀).
The numerical verification tests several nearby routes:
Paths: z = z₀ + h, z = z₀ + ih, and z = z₀ + (1+i)h.
Smaller path gaps support the reported finite limit. Larger gaps suggest branch issues, poles, or path-sensitive behavior.
How to Use This Calculator
- Select a function family that matches the complex expression you want to study.
- Enter the approach point z0 and the relevant complex coefficients.
- Adjust precision, initial step size, sample count, and tolerance for stricter verification.
- Press Calculate Limit to display the result above the form.
- Review the summary table, convergence graph, and path sample table before exporting CSV or PDF.
Frequently Asked Questions
1. What does this calculator analyze?
It studies limits of selected complex-function templates near a chosen point. It combines direct substitution, removable-singularity checks, and numerical path comparisons to support the reported limit.
2. How are complex numbers entered?
Use forms such as 3+2i, 4-5i, i, -i, 2i, or 7. Spaces are ignored, so both 1+i and 1 + i work well.
3. What happens when the denominator becomes zero?
The calculator checks whether the zero is removable. If numerator and denominator vanish with compatible orders, it resolves the limit. Otherwise, it reports that no finite limit is detected.
4. Does it handle logarithmic branch issues?
Yes. The logarithm uses the principal branch. If the inner expression approaches the branch cut or zero, the calculator warns you through the analytic note and the path comparison output.
5. Why are multiple paths compared?
A genuine complex limit must agree along every path inside the domain. Comparing real, imaginary, and diagonal routes helps reveal path-dependent behavior that direct substitution alone could miss.
6. Is the numerical check a formal proof?
No. It is strong evidence, not a proof. The analytic rules provide the main conclusion, while the path samples offer practical confirmation and help you spot unstable cases.
7. What does the Plotly graph show?
It plots the convergence metric against smaller step sizes. When a finite limit exists, the graph tracks |f(z)-L|. Otherwise, it plots |f(z)| so divergence patterns stay visible.
8. What do the CSV and PDF downloads include?
The CSV export includes the summary table and path sample table. The PDF export captures the visible result block, including the limit summary, notes, chart, and sample values.