Calculator Inputs
Plotly Graph
The graph highlights the rational function near the selected x value.
Nearby Value Table
| x | Numerator | Denominator | Rational value |
|---|---|---|---|
| 1.997 | -0.002991 | -0.003000 | 0.997 |
| 1.999 | -0.000999 | -0.001000 | 0.999 |
| 2.001 | 0.001001 | 0.001000 | 1.001 |
| 2.003 | 0.003009 | 0.003000 | 1.003 |
Example Data Table
| Case | Numerator coefficients | Denominator coefficients | Target x | Expected limit | Reason |
|---|---|---|---|---|---|
| Removable hole | 1, -3, 2 | 1, -2 | 2 | 1 | Common factor cancels. |
| Direct substitution | 2, 1 | 1, 1 | 3 | 1.75 | Denominator stays nonzero. |
| Vertical asymptote | 1 | 1, -2 | 2 | DNE | One-sided signs disagree. |
| Infinite same sign | 1 | 1, -4, 4 | 2 | +∞ | Denominator approaches zero positively. |
Formula Used
General rational limit:
limx→a P(x) / Q(x)
First evaluate P(a) and Q(a). If Q(a) ≠ 0, the limit equals P(a) / Q(a).
Removable discontinuity check:
If P(a) = 0 and Q(a) = 0, factor out (x - a).
After cancelling shared factors, evaluate the reduced expression at x = a.
Asymptotic behavior:
If Q(a) = 0 after reduction, inspect x = a ± h.
Matching one-sided trends can reveal +∞, -∞, or a missing two-sided limit.
How to Use This Calculator
- Enter numerator coefficients from highest power to constant.
- Enter denominator coefficients in the same order.
- Set the x value that the variable approaches.
- Choose two-sided, left-hand, or right-hand analysis.
- Adjust the nearby step size for more local sampling.
- Set the graph window for wider or tighter plotting.
- Press Calculate Limit to view the result above the form.
- Use the export buttons to save the result summary.
Frequently Asked Questions
1. What coefficients should I enter?
Enter coefficients from highest degree to constant. For x² - 3x + 2, type 1, -3, 2. Spaces and commas both work.
2. What happens when substitution gives 0/0?
That usually signals a removable discontinuity. The calculator tries to cancel shared factors of (x - a), then evaluates the reduced expression.
3. Why can a limit exist when the function is undefined?
A limit studies nearby behavior, not the exact function value at the point. A hole can still have a well-defined approach value.
4. When should I use left-hand or right-hand mode?
Use one-sided mode near vertical asymptotes or piecewise behavior. It shows what the function does from one side only.
5. Why does the result sometimes say DNE?
The two-sided limit does not exist when the left and right behaviors disagree, or when nearby values fail to settle toward one target.
6. How is the graph helpful?
The plot shows holes, steep growth, and sign changes near the target x value. It helps confirm whether algebra and nearby samples agree.
7. Can this tool handle high-degree polynomials?
Yes, as long as you enter valid coefficient lists. Very large degrees may become numerically sensitive, so smaller step sizes can help.
8. What do the CSV and PDF files include?
They include the main result, classification, target point, directional behavior, and the nearby value table used for interpretation.