Calculator
Formula used
For a direct absolute expression, the core continuity rule is:
For the structured forms on this page, the calculator uses these ideas:
- Single, sum, and difference modes: direct substitution works because absolute value and nonnegative powers remain continuous.
- Ratio mode with nonzero denominator: substitute directly after evaluating both absolute factors at the approach point.
- Ratio mode with a shared zero: near a common root c, |ax + b| behaves like |a|·|x - c|. This lets the tool compare powers.
When both numerator and denominator vanish in ratio mode, the calculator compares the powers:
If numerator power = denominator power, the limit becomes scale × |a₁ / a₂|power.
If numerator power < denominator power, the magnitude grows without bound.
How to use this calculator
- Choose the expression type that matches your problem.
- Enter the approach point c.
- Fill in the linear coefficients and powers for each absolute term.
- Set the nearby-check distance and graph range.
- Press Calculate Limit.
- Read the analytic result first, then verify it using the left-hand and right-hand checks.
- Inspect the table and Plotly graph for extra confirmation.
- Use the CSV and PDF buttons to export the result.
Example data table
| Example | Mode | Expression | Approach point | Expected limit |
|---|---|---|---|---|
| 1 | Single | |2x - 3| | x → 2 | 1 |
| 2 | Ratio | |x - 4| / |2x - 8| | x → 4 | 0.5 |
| 3 | Ratio | |x - 1|² / |3x - 3| | x → 1 | 0 |
| 4 | Ratio | |2x + 6| / |x + 3|² | x → -3 | ∞ |
FAQs
1) Why can absolute value limits often use direct substitution?
Absolute value is continuous. If the inner expression has a limit at the approach point, the absolute value of that limit gives the outside limit.
2) When does a finite limit fail to exist?
A finite limit can fail when the denominator tends to zero too quickly, when the expression is undefined on both sides, or when one-sided behaviors do not match.
3) Why are one-sided checks useful?
They confirm whether the function approaches the same value from the left and right. This is especially helpful near corners, holes, and denominator zeros.
4) What happens in a 0/0 absolute-value ratio?
Compare the vanishing powers. A stronger numerator gives zero, equal powers give a finite constant, and a stronger denominator gives an infinite magnitude.
5) Does a negative scale change the answer?
Yes. It flips the sign of any finite result and also changes positive infinity to negative infinity when the ratio grows without bound.
6) Can this page solve every symbolic limit problem?
It is designed for structured linear absolute-value forms with optional powers, sums, differences, and ratios. Arbitrary symbolic expressions need a full algebra system.
7) Why might the graph show gaps?
Gaps appear where the ratio becomes undefined or where values become too large for a readable plot. That usually signals a hole or vertical blow-up.
8) What precision should I choose?
Six to eight decimals is usually enough for classroom work. Use more only when nearby values change slowly or when you want tighter numeric confirmation.