Calculator Form
Example Data Table
Example function: (x^2 - 1) / (x - 1), with x → 1⁻.
| Step | x | f(x) | Trend |
|---|---|---|---|
| 1 | 0.900000 | 1.900000 | Approaching 2 from below |
| 2 | 0.980000 | 1.980000 | Closer to 2 |
| 3 | 0.996000 | 1.996000 | Very close to 2 |
| 4 | 0.999200 | 1.999200 | Stable one-sided approach |
Formula Used
Left-hand limit: limx→a⁻ f(x)
Right-hand limit: limx→a⁺ f(x)
Sample points: xn = a ± h0rn-1
Numerical estimate: median of the most recent stable sampled values
This calculator uses directional sample points that move toward the chosen approach value from one side only. It checks whether recent values cluster around a finite number, grow with one sign toward infinity, or fail to settle. The method is numerical, so it is best for estimation, interpretation, and fast verification.
How to Use This Calculator
- Enter the function using standard notation, such as
(x^2-1)/(x-1). - Set the variable name and the approach value.
- Choose whether you want the left-hand or right-hand limit.
- Adjust the initial step, decay factor, and number of samples.
- Choose decimal places and decide whether to compare the opposite side.
- Press Calculate Limit to view the result above the form.
- Review the interpretation, sample tables, and export options.
FAQs
1. What is a one sided limit?
A one sided limit studies the behavior of a function as the variable approaches a value from only the left side or only the right side.
2. Can a limit exist if the function is undefined there?
Yes. A limit depends on nearby behavior, not the actual value at the exact point. Removable discontinuities often have this pattern.
3. What does an infinite classification mean?
It means the sampled values keep growing in magnitude with a consistent sign as they move toward the approach point from the chosen side.
4. Why compare the opposite side too?
Comparing both sides helps determine whether the full two-sided limit likely exists. Matching side estimates usually indicate agreement.
5. Which functions can I enter?
You can enter algebraic, exponential, logarithmic, and many trigonometric expressions using the supported functions shown below the form.
6. Why should I use explicit multiplication?
Explicit multiplication reduces ambiguity. Writing 2*x is safer than 2x, especially in more complex expressions.
7. Does this tool give a symbolic proof?
No. It provides a strong numerical estimate and interpretation. For formal proofs, use algebraic simplification, limit laws, or calculus theorems.
8. How do I choose the decay factor?
A value like 0.2 or 0.1 works well in most cases. Smaller factors move faster toward the target and can reveal sharper local behavior.