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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

limitx yx2+y2,x=0,y=0

12..12

(1)

limitx+y2x2x y+y2,x=0,y=0

0..4

(2)

Support for functions containing abs or radicals

fx+|y|x2+y2:

limitf,x=0,y=0

undefined

(3)

Why?

fy|xx=0

yy2

(4)

simplify assuming real

1

(5)

fy|xx=y

y2y2

(6)

simplify assuming real

2

(7)

limitx2+y2x2x y+y2,x=0,y=0

0

(8)

limit(|x|+yx4+3 x2y2+y4,x=0,y=0)

(9)

Support for functions in more than 2 variables

limitexpx3+y3+z31sinx2+y2+z2,x=0,y=0,z=0

0

(10)

f  x2+y2+z2x+y+z:

limitf,x=0,y=0,z=0

undefined

(11)

Why?

fy|xy=0,z=0

x

(12)

limit,x=0

0

(13)

fy|xy=0,z=x+x4

x2+x4x2x4

(14)

limit,x=0

(15)

fx2+y2+z2+x y zx4+2 x2z2+z4+y2y3:

limitf,x=0,y=0,z=0

1

(16)

Why?

simplifyf assuming real

xyz+x2+y2+z2x2+z2+y2y3

(17)

fx4+y4+z4x2+y2+z2:

limitf,x=0,y=0,z=0

undefined

(18)

Why?

fy|xy=0,z=0

x4x2

(19)

simplify assuming real

1

(20)

fy|xy=x,z=x

3x43x2

(21)

simplify assuming real

33

(22)

limit(x+y2y z+z2+|x y z|x2+x y+y2+y z+z2x3+y3z3,x=0,y=0,z=0)

(23)

f  x y+x z+y z1cos x2+y2+z2:

limitf,x=0,y=0,z=0

undefined

(24)

Why?

fy|xy=0,z=0

0

(25)

fy|xy=x,z=x

3x21cos3x2

(26)

limit,x=0

(27)

See Also

updates/Maple17/BivariateLimits, updates/Maple2015/BivariateLimits, updates/Maple2017/BivariateLimits, limit, limit/multi - multidimensional limits