Function Repository Resource:

# Asymptotes

Compute the asymptotes to a given curve in two dimensions

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["Asymptotes"][expr,x,y] finds the asymptotes of the expression expr in terms of independent variable x and dependent variable y. ResourceFunction["Asymptotes"][expr,x,y,type] finds the asymptotes of expr whose type matches the string argument type.

## Details

When called with three arguments, ResourceFunction["Asymptotes"] returns an association whose keys are among the following asymptote types: "Horizontal", "Vertical", "Oblique", "Parabolic" or "Other", and whose values are lists of asymptotes of the given type.
If a type is given as a fourth argument to ResourceFunction["Asymptotes"], the result is a list of asymptotes of the specified type. Acceptable values for type are "Horizontal", "Vertical", "Oblique", "Parabolic", "Other" or All.
The first argument to ResourceFunction["Asymptotes"] can be either an expression involving x alone, specifying the curve y(x) =expr, or an equation involving both x and y.
ResourceFunction["Asymptotes"] takes the option "SingleStepTimeConstraint", which specifies the maximum time (in seconds) to spend on an individual internal step of the calculation. The default value of "SingleStepTimeConstraint" is 5.

## Examples

### Basic Examples (9)

Compute the asymptotes of a decaying exponential:

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Compute the asymptotes of a hyperbola:

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Compute the asymptotes of a rational function:

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Compute only the oblique asymptotes of the previous function:

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Compute the asymptotes of a periodic function:

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Compute the asymptotes of an algebraic function:

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Compute a list of all asymptotes of the previous function:

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Compute the asymptotes of another algebraic function:

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Compute the asymptotes of a transcendental function:

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### Scope (3)

Implicitly defined curves can be specified using an expression with head Equal in the first argument:

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In certain cases, parabolic asymptotes may be returned:

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Asymptotes found that are neither linear or parabolic are classified as "Other":

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## Publisher

Wolfram|Alpha Math Team

## Version History

• 6.0.0 – 23 March 2023
• 5.1.0 – 11 May 2021
• 5.0.0 – 24 January 2020
• 4.0.0 – 06 January 2020
• 3.0.0 – 06 September 2019
• 2.0.0 – 12 June 2019
• 1.0.0 – 22 January 2019