Result
Enter values and press Calculate Limit.
The calculator checks direct substitution, nearby sample points, one-sided behavior, and a plotted curve.
Calculator Input
Set any radical coefficient to 0 if you want to remove that term.
Plotly Graph
The graph uses valid real-domain points only.
Nearby Sample Points
| Side | h | x | f(x) |
|---|---|---|---|
| Submit the form to generate nearby sample points. | |||
Example Data Table
Example expression: (√(x + 9) - 5) / (x - 16), with x → 16.
| x | f(x) |
|---|---|
| 15.9 | 0.100100200501 |
| 15.99 | 0.100010002001 |
| 16.01 | 0.099990001999 |
| 16.1 | 0.099900199501 |
The values move toward 0.1 from both sides, so the limit is 0.1.
About This Calculator
Limits of radical functions often look simple at first, but the square root restrictions create domain conditions that can change the result. A function may appear harmless near a point, yet one side can leave the real domain, or a denominator can collapse to zero. This calculator is designed for those situations.
The form accepts two radical terms and one linear term in the numerator, plus the same structure in the denominator. That gives enough flexibility for many textbook and classroom limit problems. You can test direct substitution, removable discontinuities, one-sided domain breaks, and finite or infinite limit behavior.
After submission, the page places the result directly below the header and above the form. It also builds a nearby sample table and a Plotly graph. The sample values help you inspect left-hand and right-hand behavior numerically. The graph then makes the same trend easier to see over a wider interval.
The export tools help with practice, documentation, and sharing. CSV is useful when you want the nearby points in a spreadsheet. PDF is useful for saving the expression, result summary, and numeric table in one report. The included example table also shows how a classic radical limit approaches the same value from both sides.
Formula Used
The calculator evaluates a function of the form:
It applies these ideas:
- Direct substitution when every used radicand stays nonnegative and the denominator is nonzero.
- One-sided numerical sampling when direct substitution fails or becomes inconclusive.
- Real-domain checking, since every active square root must satisfy inside value ≥ 0.
- Continuity logic, because radical expressions are continuous where they are defined.
A common rationalization identity for radical limits is:
That identity helps with classic forms such as:
By multiplying by the conjugate, that expression becomes:
Then the limit at x → 16 is 1 / 10.
How to Use This Calculator
- Enter the limit point where x approaches a specific value.
- Fill the numerator coefficients A through H.
- Fill the denominator coefficients P through W.
- Set any unused radical coefficient to 0.
- Choose a graph span and a graph point count.
- Press Calculate Limit.
- Read the result summary placed above the form.
- Review the nearby sample table and the graph.
- Download CSV or PDF if you want a saved copy.
FAQs
1. What kinds of radical limits can this page handle?
It handles many real-valued radical expressions built from up to two square root terms and one linear term in both numerator and denominator.
2. When does direct substitution work?
Direct substitution works when every active radicand stays nonnegative at the limit point and the denominator does not become zero there.
3. Why are one-sided limits important for radicals?
Radical expressions can exist on one side of a point and fail on the other. One-sided checks reveal whether the full two-sided limit is possible.
4. Why might the calculator report that the limit does not appear to exist?
The two sides may approach different numbers, one side may leave the real domain, or the function may grow without matching behavior from both directions.
5. What happens if I set a radical coefficient to 0?
That radical term is ignored. Its inside expression no longer affects the domain or the function value in the current calculation.
6. Does this calculator prove a symbolic answer?
It is mainly a numeric and structural tool. It supports reasoning well, but formal symbolic proofs should still be written separately when required.
7. Why can the graph have gaps?
Gaps appear where the expression is outside the real domain or where the denominator becomes zero, so those points are skipped intentionally.
8. What do the export buttons save?
CSV saves the nearby sample table. PDF saves the expression, result summary, and the same nearby points in a compact report.