Calculator
Example Data Table
This sample uses (x^2 - 4) / (x - 2) as x approaches 2.
| x | x^2 - 4 | x - 2 | Ratio |
|---|---|---|---|
| 1.9 | -0.39 | -0.1 | 3.9 |
| 1.99 | -0.0399 | -0.01 | 3.99 |
| 2.01 | 0.0401 | 0.01 | 4.01 |
| 2.1 | 0.41 | 0.1 | 4.1 |
Formula Used
For a finite point x = a, first test direct substitution: Limit = P(a) / Q(a) when Q(a) is not zero.
If both P(a) and Q(a) are zero, compare how many times each polynomial contains the factor (x - a). If the zero orders match, the limit equals the ratio of the first nonzero derivative values.
For x approaching positive or negative infinity, compare polynomial degrees. Lower numerator degree gives zero. Equal degrees give the leading coefficient ratio. Higher numerator degree gives positive or negative infinity, depending on the signs and parity.
How to Use This Calculator
- Enter numerator coefficients from x^3 down to the constant term.
- Enter denominator coefficients in the same order.
- Select whether x approaches a finite value, positive infinity, or negative infinity.
- For a finite limit, enter the approach value a.
- Set graph start, graph end, and sample points for the Plotly chart.
- Click Calculate Limit to show the result above the form.
- Review the nearby value table and graph to confirm the trend.
- Use the CSV and PDF buttons to export the current result.
FAQs
1. What does a rational limit mean?
A rational limit studies the behavior of one polynomial divided by another as x approaches a chosen value or infinity. It shows the value or trend the expression approaches, even if the function is undefined at that exact point.
2. When can I use direct substitution?
Use direct substitution when the denominator does not become zero at the approach value. In that case, the rational function is continuous there, so the limit equals the ordinary function value.
3. What if I get 0 over 0?
A 0/0 form means more work is needed. For rational functions, you usually compare repeated factors or repeated derivatives until the first nonzero terms appear. Then you evaluate the resulting simpler ratio.
4. Why does the calculator show one-sided behavior?
Some rational limits do not match from both sides. A denominator with an odd extra factor can send the left side toward one infinity and the right side toward the opposite infinity. Then the two-sided limit does not exist.
5. How are infinity limits decided?
Infinity limits depend mainly on the highest powers. Compare the degrees first, then the leading coefficients. That tells you whether the expression approaches zero, a constant ratio, or an unbounded positive or negative value.
6. Why is the graph broken near some x-values?
The graph breaks where the denominator becomes zero or where values grow too large to plot clearly. Those gaps help reveal vertical asymptotes and undefined points in the rational expression.
7. Can this calculator handle removable discontinuities?
Yes. If numerator and denominator share the same factor, the calculator compares their zero orders and can return the finite limit after cancellation behavior is detected through derivative-based checking.
8. What should I export with CSV or PDF?
Use CSV when you want the numerical table in spreadsheet form. Use PDF when you want a compact report with the equation, chosen approach, detected method, and the nearby evaluation table.