Description
Return rational leading sign at a supplied infinity direction.
Local polynomial and rational-function behavior depends on the order of zeros and poles and on the sign of the leading term. Canceling equal local orders differs from evaluating an undefined original expression; a zero placeholder for a pole must be read with its finite/defined flag.
Return rational leading sign at a supplied infinity direction.
When to use this tool
Use Rational Leading Term Sign when checking the stated local order, leading behavior or limit convention.
How to use the tool and how it works
- Enter numeratorLeading, denominatorLeading, approachPositiveInfinity, numeratorDegree, denominatorDegree. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read leadingSign, degreeDifference. Use the output convention below.
Return rational leading sign at a supplied infinity direction. Require nonzero leading coefficients and nonnegative whole degrees; degreeDifference = numerator-denominator, sign combines coefficient signs and negative-infinity parity.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| numeratorLeading | Numerator Leading | Required. Finite number. |
| denominatorLeading | Denominator Leading | Required. Finite number. |
| approachPositiveInfinity | Approach Positive Infinity | Required. Boolean. |
| numeratorDegree | Numerator Degree | Required. Safe integer; minimum 0. |
| denominatorDegree | Denominator Degree | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| leadingSign | Leading Sign | Required. Safe integer. |
| degreeDifference | Degree Difference | Required. Safe integer. |
{
"input": {
"numeratorLeading": 2,
"denominatorLeading": -3,
"approachPositiveInfinity": true,
"numeratorDegree": 3,
"denominatorDegree": 1
},
"output": {
"leadingSign": -1,
"degreeDifference": 2
}
}Limitations and assumptions
- This is the sign of the leading asymptotic term, not necessarily a finite limit or proof that supplied degrees/coefficients describe a valid function.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the complete function, original domain and an appropriate symbolic limit or asymptotic analysis for a full mathematical conclusion.
References
Similar or alternative tools
- Rational Function Infinity Order
Classify rational order at infinity from supplied degrees.
- Algebra: Polynomial Leading Nonzero Coefficient Tool
Extract leading coefficient and degree from ascending-power coefficients using the existing normalization core; zero polynomial has degree -1.