Bivariate Limits - Maple Programming Help

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Bivariate Limits

The limit command has been enhanced for the case of limits of bivariate rational functions with non-isolated singularities. Many such limits that could not be determined previously are now computable. If the limit exists in such a situation, it is either +∞ or ∞. Maple can also determine if the limit does not exist, and then returns undefined.

In Maple 18, all the following limit calls would return unevaluated, but they can now be computed in Maple 2015.

fx yx+y:

limitf,x=0,y=0

undefined

(1)

g4 x yxy2:

limitg,x=0,y=0

undefined

(2)

hx4x2y2xy4:

limith,x=0,y=0

(3)

Let us plot these three functions in the neighborhood of the origin.

In the first example, f  tends to +∞ on one side of the singularity y=x and to ∞ on the other side (as shown in the following plot). Therefore, the limit at the origin does not exist.

pf1plot3df, x=0.1..0.1,y=0.1..x1e−10,axes=boxed,view=10..10:

pf2plot3df, x=0.1..0.1,y=x+1e−10..0.1,axes=boxed,view=10..10:

plots:-displaypf1,pf2

Now, consider the second example.

splots:-spacecurvex,x,1,x=0.1..0.1,color=red,thickness=3:pgplot3dg,x=0.1..0.1,y=0.1..0.1,axes=boxed,view=10..100,numpoints=40000:plots:-displays,pg

g tends to +∞ close to the singularity y=x. However, along the anti-diagonal y=x, the limit is finite:

evalg,y=x

−1

(4)

Thus, g does not have a limit at the origin. In fact, any number1 can occur, namely, as the limit along the ray y&equals;a x for 1a<1:

evalg&comma;y&equals;a x

4x2aax+x2

(5)

limit&comma;x&equals;0

4aa12

(6)

plot&comma;a&equals;3..1.1&comma;view&equals;2..10

 

In the last example, h tends to &infin; on both sides of the singularity y&equals;x.

plot3dh&comma;x&equals;0.1..0.1&comma;y&equals;0.1..0.1&comma;axes&equals;boxed&comma;view&equals;100000..1

However, in this case the limit along any ray y&equals;a x with a1 is &infin; as well.

evalh&comma;y&equals;a x

a2x2+x4x2ax+x4

(7)

limit&comma;x&equals;0

signuma2+1a14

(8)

factor

signuma2+1a14

(9)

simplify assuming a<1&semi;simplify assuming a&gt;1

(10)

You can prove that the limit exists and is &infin; for any curve approaching the origin by using the theory of Lagrange multipliers. The extremal values (maxima and minima) of the function h  on the circle, for a fixed radius r, satisfy the condition that the gradient of the function and the gradient of the constraint equation of the circle are parallel:

Cx2&plus;y2r2&colon;

withVectorCalculus&colon;

dfnormal~Jacobianh&comma;x&comma;y

22x3yx2xy2y2xy522x42x2xyy2xy5

(11)

dCJacobianC&comma;x&comma;y

2x2y

(12)

eqfactornumernormaldf1&comma;1dC1&comma;2df1&comma;2dC1&comma;1

eq8x3xyx2+y2

(13)

Thus, the local maximum and minimum values of f  on C occur when both C&equals;0 and eq&equals;0. For the bivariate limit, this means that it is sufficient to consider only those critical paths satisfying eq&equals;0. However, you also need to consider the global suprema and infima, which may occur close to the singularity y&equals;x.
In the example, the factor x2&plus;y2 of eq does not admit any real paths, so there is only one critical path given by x3xy&equals;0 (or, equivalently, y&equals;x&plus;x3).

normalevalh&comma;y&equals;x&plus;x3

x21x2x223

(14)

limit&comma;x&equals;0

(15)

In order to certify the limit close to the singularity y&equals;x as well, you cannot take the limit along the singularity. Instead, consider two curves that approach the singularity closely from the top and from the bottom, respectively:

c1y&equals;x&plus;x2

c1y=x2+x

(16)

c2y&equals;xx2

c2y=x2+x

(17)

plotx&comma;rhsc1&comma;rhsc2&comma;x&plus;x3&comma;x&equals;0.5..0.5&comma;        legend&equals;singular path&comma;c1&comma;c2&comma;critical path

normalevalh&comma;c1

2x+1x6

(18)

limit&comma;x&equals;0

(19)

normalevalh&comma;c2

2x1x6

(20)

limit&comma;x&equals;0

(21)

See Also

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