> > > > > > > > > > (1) (1) > > > > (2) (2) > > > > (3) (3) 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 . In Maple 18, all the following limit calls would return unevaluated, but they can now be computed in Maple 2015. undefined undefined Let us plot these three functions in the neighborhood of the origin. In the first example, tends to on one side of the singularity and to on the other side (as shown in the following plot). Therefore, the limit at the origin does not exist.
Bivariate LimitsThe 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 canalso determine if the limit
does not exist, and then returns .
In Maple 18, all the following limit calls would return
unevaluated, but they can now be computed in Maple 2015.
undefined
undefined
Let us plot these three functions in the neighborhood of the
origin.
In the first example, tends to on one side of the singularity
and to on the other side (as shown in the following plot).
Therefore, the limit at the origin does not exist.
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Now, consider the second example.
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(4)(4)
(5)(5)
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(6)(6)
tends to close to the singularity . However, along the
anti-diagonal , the limitis finite:
Thus, does not have a limit at the origin. In fact, any number
can occur, namely, as the limit along the ray for :
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In the last example, tends to on both sides of the singularity
.
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(10)(10)
(8)(8)
(9)(9)
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(7)(7)
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However, in this case the limit along any ray with is as
well.
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(17)(17)
(13)(13)
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(11)(11)
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(16)(16)
(12)(12)
(15)(15)
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(14)(14)
You can prove that the limit exists and is for any curve
approaching the origin by using the theory of Lagrange multipliers.
The extremal values (maxima and minima) of the function on the
circle, for a fixed radius , satisfy the condition that the
gradient of the function and the gradient of the constraint
equation of the circle are parallel:
Thus, the local maximum and minimum values of on occur when both
and For the bivariate limit, this means that it is sufficient to
consider only those critical paths satisfying . However, you also
need to consider the global suprema and infima, which may occur
close to the singularity .In the example, the factor of does not
admit any real paths, so there is only one critical path given by
(or, equivalently, ).
In order to certify the limit close to the singularity 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:
