Multivariate Limits - New Features in Maple 2019 - Maplesoft

What's New in Maple 2019

Pricing & Purchase
Upgrade to Maple 2019
Back to What's New Overview

Multivariate Limits



The limit command in Maple 2019 has been enhanced for the case of limits of quotients of multivariate functions: 

Many such limits that could not be determined previously are now computable, including all of the following examples. 

Returning ranges instead of undefined in the bivariate case 

> limit(`/`(`*`(x, `*`(y)), `*`(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2))))), {x = 0, y = 0})
 

-`/`(1, 2) .. `/`(1, 2)
 

> limit(`/`(`*`(`^`(`+`(x, y), 2)), `*`(`+`(`*`(`^`(x, 2)), `-`(`*`(x, `*`(y))), `*`(`^`(y, 2))))), {x = 0, y = 0})
 

0 .. 4
 

Support for functions containing abs or radicals 

> f := `/`(`*`(`+`(abs(x), abs(y))), `*`(sqrt(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)))))); -1
 

> limit(f, {x = 0, y = 0})
 

undefined
 

Why? 

> eval(f, x = 0)
 

`/`(`*`(abs(y)), `*`(`^`(`*`(`^`(y, 2)), `/`(1, 2))))
 

> `assuming`([simplify(`/`(`*`(abs(y)), `*`(`^`(`*`(`^`(y, 2)), `/`(1, 2)))))], [real])
 

1
 

> eval(f, x = y)
 

`/`(`*`(abs(y), `*`(`^`(2, `/`(1, 2)))), `*`(`^`(`*`(`^`(y, 2)), `/`(1, 2))))
 

> `assuming`([simplify(`/`(`*`(abs(y), `*`(`^`(2, `/`(1, 2)))), `*`(`^`(`*`(`^`(y, 2)), `/`(1, 2)))))], [real])
 

`*`(`^`(2, `/`(1, 2)))
 

> limit(`/`(`*`(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)))), `*`(sqrt(`+`(`*`(`^`(x, 2)), `-`(`*`(x, `*`(y))), `*`(`^`(y, 2)))))), {x = 0, y = 0})
 

0
 

> limit(`/`(`*`(`+`(abs(x), abs(y))), `*`(`+`(`*`(`^`(x, 4)), `*`(3, `*`(`^`(x, 2), `*`(`^`(y, 2)))), `*`(`^`(y, 4))))), {x = 0, y = 0})
 

infinity
 

Support for functions in more than 2 variables 

> limit(`/`(`*`(`+`(exp(`+`(`*`(`^`(x, 3)), `*`(`^`(y, 3)), `*`(`^`(z, 3)))), `-`(1))), `*`(sin(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2)))))), {x = 0, y = 0, z = 0})
 

0
 

> f := `/`(`*`(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2)))), `*`(`+`(x, y, z))); -1
 

> limit(f, {x = 0, y = 0, z = 0})
 

undefined
 

Why? 

> eval(f, [y = 0, z = 0])
 

x
 

> limit(x, x = 0)
 

0
 

> eval(f, [y = 0, z = `+`(`*`(`^`(x, 4)), `-`(x))])
 

`/`(`*`(`+`(`*`(`^`(x, 2)), `*`(`^`(`+`(`*`(`^`(x, 4)), `-`(x)), 2)))), `*`(`^`(x, 4)))
 

> limit(`/`(`*`(`+`(`*`(`^`(x, 2)), `*`(`^`(`+`(`*`(`^`(x, 4)), `-`(x)), 2)))), `*`(`^`(x, 4))), x = 0)
 

infinity
 

> f := `/`(`*`(`+`(`*`(x, `*`(y, `*`(z))), `*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2)))), `*`(`+`(sqrt(`+`(`*`(`^`(x, 4)), `*`(2, `*`(`^`(x, 2), `*`(`^`(z, 2)))), `*`(`^`(z, 4)))), `*`(`^`(y, 2)), `-...
 

> limit(f, {x = 0, y = 0, z = 0})
 

1
 

Why? 

> `assuming`([simplify(f)], [real])
 

`/`(`*`(`+`(`*`(x, `*`(y, `*`(z))), `*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2)))), `*`(`+`(`*`(`^`(x, 2)), `*`(`^`(z, 2)), `*`(`^`(y, 2)), `-`(`*`(`^`(abs(y), 3))))))
 

> f := `/`(`*`(sqrt(`+`(`*`(`^`(x, 4)), `*`(`^`(y, 4)), `*`(`^`(z, 4))))), `*`(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2))))); -1
 

> limit(f, {x = 0, y = 0, z = 0})
 

undefined
 

Why? 

> eval(f, [y = 0, z = 0])
 

`/`(`*`(`^`(`*`(`^`(x, 4)), `/`(1, 2))), `*`(`^`(x, 2)))
 

> `assuming`([simplify(`/`(`*`(`^`(`*`(`^`(x, 4)), `/`(1, 2))), `*`(`^`(x, 2))))], [real])
 

1
 

> eval(f, [y = x, z = x])
 

`+`(`/`(`*`(`/`(1, 3), `*`(`^`(3, `/`(1, 2)), `*`(`^`(`*`(`^`(x, 4)), `/`(1, 2))))), `*`(`^`(x, 2))))
 

> `assuming`([simplify(`+`(`/`(`*`(`/`(1, 3), `*`(`^`(3, `/`(1, 2)), `*`(`^`(`*`(`^`(x, 4)), `/`(1, 2))))), `*`(`^`(x, 2)))))], [real])
 

`+`(`*`(`/`(1, 3), `*`(`^`(3, `/`(1, 2)))))
 

> limit(`/`(`*`(`+`(abs(x), sqrt(`+`(`*`(`^`(y, 2)), `-`(`*`(y, `*`(z))), `*`(`^`(z, 2)))), abs(`*`(x, `*`(y, `*`(z)))))), `*`(`+`(`-`(`*`(`^`(x, 3))), `*`(`^`(y, 3)), `-`(`*`(`^`(z, 3))), `*`(`^`(x, 2)...
 

infinity
 

> f := `/`(`*`(`+`(`*`(x, `*`(y)), `*`(x, `*`(z)), `*`(y, `*`(z)))), `*`(`+`(1, `-`(cos(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)), `*`(`^`(z, 2)))))))); -1
 

> limit(f, {x = 0, y = 0, z = 0})
 

undefined
 

Why? 

> eval(f, [y = 0, z = 0])
 

0
 

> eval(f, [y = x, z = x])
 

`+`(`/`(`*`(3, `*`(`^`(x, 2))), `*`(`+`(1, `-`(cos(`+`(`*`(3, `*`(`^`(x, 2))))))))))
 

> limit(`+`(`/`(`*`(3, `*`(`^`(x, 2))), `*`(`+`(1, `-`(cos(`+`(`*`(3, `*`(`^`(x, 2)))))))))), x = 0)
 

infinity
 



Back to What's New Overview
Prêt(e) pour l'étape suivante ?
Discutez aujourd'hui avec nos spécialistes produits pour évaluation gratuite de Maple.

*Ce programme d'évaluation n'est proposé actuellement ni aux étudiants ni aux particuliers.


Achetez:

 

Licences mono-utilisateurs

Licences multi-utilisateurs

Effectuez la mise à jour vers Maple 2025