Description
Return a rational-function local limit from supplied orders and leading terms.
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 a rational-function local limit from supplied orders and leading terms.
When to use this tool
Use Rational Pole Removal Limit Coefficient when checking the stated local order, leading behavior or limit convention.
How to use the tool and how it works
- Enter numeratorOrder, denominatorOrder, numeratorLeading, denominatorLeading. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read limitDefined, limit, residualOrder. Inspect limitDefined before interpreting the associated values.
Return a rational-function local limit from supplied orders and leading terms. Require nonnegative whole orders and nonzero local leading coefficients; residualOrder = numerator-denominator, limitDefined means residual >= 0, limit is coefficient ratio when equal or zero otherwise.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| numeratorOrder | Numerator Order | Required. Safe integer; minimum 0. |
| denominatorOrder | Denominator Order | Required. Safe integer; minimum 0. |
| numeratorLeading | Numerator Leading | Required. Finite number. |
| denominatorLeading | Denominator Leading | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| limitDefined | Limit Defined | Required. Boolean. |
| limit | Zero placeholder when residualOrder<0 and limitDefined=false; otherwise uses the stated local-order rule. | Required. Finite number. |
| residualOrder | Residual Order | Required. Safe integer. |
{
"input": {
"numeratorOrder": 2,
"denominatorOrder": 2,
"numeratorLeading": 6,
"denominatorLeading": 3
},
"output": {
"limitDefined": true,
"limit": 2,
"residualOrder": 0
}
}Limitations and assumptions
- When denominator order is larger, zero limit is a placeholder with limitDefined false; orders, smooth expansions and cancellation are supplied rather than derived.
- 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 Pole Order From Orders
Return rational pole order and removable flag.
- Rational Function Zero Order From Orders
Return rational zero order and a pole flag.