Calculator Inputs
Choose a trigonometric limit form, set the coefficients, then calculate the exact and decimal limit with supporting steps.
Example Data Table
These examples show common presets and the expected limit values.
| Preset | k | m | a | Unit | Expected Limit |
|---|---|---|---|---|---|
| sin(k(x-a)) / (m(x-a)) | 3 | 2 | 0 | Radians | 1.5 |
| tan(k(x-a)) / (m(x-a)) | 5 | 4 | 1 | Radians | 1.25 |
| (1 - cos(k(x-a))) / (m(x-a)^2) | 4 | 2 | 0 | Radians | 4 |
| sin(k(x-a)) / tan(m(x-a)) | 6 | 3 | -2 | Radians | 2 |
| (1 - cos(k(x-a))) / sin²(m(x-a)) | 2 | 4 | 0 | Radians | 0.125 |
Formula Used
- sin(u) / u → 1 as u → 0.
- tan(u) / u → 1 as u → 0.
- 1 - cos(u) ~ u² / 2 near zero.
- sin(u) ~ u and tan(u) ~ u for small angles.
- tan(u) - sin(u) ~ u³ / 2 near zero.
- For degree-based inputs, angles are converted using u = (π/180) × angle.
- Shifted limits use h = x - a, so every selected preset is evaluated as h → 0.
How to Use This Calculator
- Choose the trigonometric limit preset matching your expression.
- Enter values for k, m, and the target point a.
- Pick radians or degrees, then choose two-sided, left-hand, or right-hand approach.
- Set the graph window, nearby sample count, and decimal precision.
- Press Calculate Limit to show the result above the form.
- Review the exact form, decimal value, steps, nearby values, and graph.
- Use the export buttons to download a CSV sheet or a compact PDF summary.
FAQs
1. What does this calculator solve?
It solves selected trigonometric limits built from sine, cosine, tangent, and mixed small-angle forms. It also shows exact expressions, decimal results, nearby function values, and a graph around the target point.
2. Why do many trigonometric limits rely on radians?
Standard small-angle identities are naturally defined in radians. When you choose degrees here, the calculator converts angle arguments internally, so the final limit remains mathematically consistent.
3. What is the meaning of the target point a?
The point a is the value that x approaches. The calculator rewrites each selected form using h = x - a, which turns the problem into a small-angle limit near zero.
4. Can I study one-sided limits here?
Yes. You can choose left-hand, right-hand, or two-sided approach. The analytical result matches the chosen preset, while the nearby sample table reflects the selected direction.
5. Why does the graph skip the exact target point?
Many limit expressions are undefined exactly at the target point because they produce forms like 0/0. The graph therefore plots values around the point and marks the computed limit separately.
6. What does the nearby values table show?
It lists x-values close to the target point, the offset from that point, and the evaluated function values. This helps you see the numerical approach toward the theoretical limit.
7. Which preset is best for mixed expressions?
Use the mixed presets when both sine and tangent appear, or when cosine is compared against squared sine. Those presets apply multiple standard identities in one calculation.
8. Can I download the results for assignments?
Yes. After calculating, use the CSV button for spreadsheet work or the PDF button for a concise printable summary containing the expression, result, and main steps.