| Scenario | Expression or Definition | Point a | Expected Outcome |
|---|---|---|---|
| Removable Discontinuity | (x^2 - 1) / (x - 1) | 1 | Limit exists and equals 2, but f(1) is undefined. |
| Continuous Polynomial | x^3 - 4*x + 1 | 2 | Left, right, and function value match. |
| Jump Discontinuity | Left: x+2, Right: 3-x, f(1)=3 | 1 | Left and right limits differ. |
Continuity condition at x = a:
- The function value f(a) exists.
- The limit lim x→a f(x) exists.
- The two quantities are equal: lim x→a f(x) = f(a).
One-sided numerical checks:
L ≈ f(a - h), R ≈ f(a + h)
If |L - R| ≤ tolerance, the limit is treated as existing numerically.
If |Limit - f(a)| ≤ tolerance, the function is treated as continuous at a.
- Select Single Expression for one formula or Piecewise at a Point for separate left and right formulas.
- Enter the expression, test point, tolerance, and sampling step.
- Press Submit to place the result above the form and below the header.
- Review the classification, limit values, and nearby table.
- Export the result summary as CSV or PDF when needed.
Understanding Continuity Checks
A continuity limit calculator checks whether a function behaves smoothly near a chosen point. It estimates the left hand limit, right hand limit, and function value, then compares them within a tolerance. This helps with rational, radical, logarithmic, trigonometric, and piecewise expressions. Students can confirm whether a point is continuous instead of relying only on inspection. It also reduces manual substitution mistakes when evaluating tricky expressions near excluded points.
Why One Sided Limits Matter
Continuity at x = a needs agreement from both directions. If the left hand estimate differs from the right hand estimate, the two sided limit fails. That happens in jump discontinuities and many piecewise rules. Sampling values at a − h and a + h gives a practical numerical picture of how the function approaches the point from each side.
Interpreting Removable Cases
A removable discontinuity appears when the common limit exists but the function value is missing or incorrect. For example, (x^2 − 1)/(x − 1) near x = 1 approaches 2, even though the original expression is undefined there. The calculator identifies this by matching one sided estimates while detecting that f(a) does not agree.
Tolerance and Sampling Effects
Numerical results depend on the sampling step and tolerance. A smaller h usually improves local estimation, but extremely tiny steps can increase rounding noise. Tolerance decides when two values are treated as equal. In typical study use, moderate settings such as h = 0.1 or 0.01 with a small tolerance give stable classifications for most textbook examples.
Reading the Data Table and Graph
The nearby values table lists sampled x values around the selected point and shows the related outputs. The graph then displays the same neighborhood visually. Smooth alignment suggests local continuity, while gaps, jumps, or sharp divergence suggest discontinuity. Using both views together helps learners connect numerical limits with graph behavior and strengthens interpretation during revision.
Where This Tool Helps Most
This tool supports algebra, precalculus, and calculus practice. Students can verify homework answers, while teachers can demonstrate the formal continuity conditions with clear numerical evidence. It is also useful for timed exam preparation because classification becomes faster when left, right, and point values appear together. Exports make it easier to save results for notes or review. In applied modeling, continuity checks can also highlight unrealistic breaks in formulas used for cost, motion, or response data over time.
What does this calculator test?
It estimates left hand and right hand limits, checks the function value at the chosen point, and classifies the point as continuous or discontinuous.
Can it analyze piecewise functions?
Yes. Use piecewise mode to enter separate left and right expressions and the defined value at the point being tested.
Why is tolerance important?
Tolerance controls when two numerical estimates are treated as equal. It helps manage rounding effects during approximate limit evaluation.
What does a removable discontinuity mean?
It means the common limit exists, but the function value is either missing or does not match that limit at the chosen point.
How does the graph help?
The graph shows sampled points near the target value, making holes, jumps, or smooth local behavior easier to interpret visually.
Is this suitable for exam preparation?
Yes. It speeds up classification practice and helps students connect continuity rules with numerical evidence and local graph behavior.