# Limits Question

How can one prove that:

lim (x,y)->(0,0)

(x^4+y^4)
---------
(x^2+y^2)

= 0

I keep getting 0/0 no matter what I do to the equation. Anyone have any pointers?

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Staff Emeritus
Gold Member
Apply the epsilon-delta definition of a limit, or try a clever substitution.

Hurkyl

Thank you for the quick response. I'll go and try it out now.

Homework Helper
One of the difficulties with limits in more than one variable is that there are an infinite number of different ways to "approach" the target point. In order for the limit to exist, the result must be the same along any path.

Since the "epsilon-delta" definition Hurkyl mentioned used the distance from the target point the best way to use it is to change to polar coordinates so that r measures the distance from from (0,0).

Note that x= r cos([theta]) and y= r sin([theta]) so that
x<sup>2</sup>+ y<sup>2</sup>= r<sup>2</sup> and
x<sup>4</sup>+ y<sup>4</sup>= r<sup>4</sup>(cos<sup>4</sup>([theta])+sin<sup>4</sup>([theta]).

That should make it easy.

Homework Helper
Oops, wrong forum, wrong symbols. Well, the math is still correct.

Loren Booda
Would using L'Hospital's rule (successive differentiations of numerator and denominator) be cheating?

Homework Helper
Since this is a function of two variables, HOW, exactly, would you apply L'Hospital's rule?

Thanks HallsOfIvy. That was much easier to work with

Loren Booda
HallsofIvy
Since this is a function of two variables, HOW, exactly, would you apply L'Hospital's rule?
(d2N/dxNdyN)((x^4+y^4)/(x^2+y^2)),

where the derivatives are partial.

ObsessiveMathsFreak
the proof is as follows.

lim(x,y)->(0,0)

x^4 + y^4
---------
x^2 + y^2

=

(x^2 + i*y^2)*(x^2 - i*y^2)
---------------------------
x^2 + y^2

=

x^2 - i*y^2

as x and y go to zero this value approches 0.

Not that when they are zero the value of the function is NOT zero.

Loren Booda
ObsessiveMathsFreak
Not that when they are zero the value of the function is NOT zero.
Is that value then undefined?

bogdan
Why

(x^2 + i*y^2)*(x^2 - i*y^2)
---------------------------
x^2 + y^2

=

x^2 - i*y^2

?

x^2 + i*y^2<>x^2 + y^2

Try this way :

(x^4+y^4)/(x^2+y^2)=((x^2+y^2)^2-2*x^2*y^2)/(x^2+y^2)=
=1-2*x^2*y^2/(x^2+y^2);
Now lim (x^2+y^2)-2*x^2*y^2/(x^2+y^2) = lim (x^2+y^2) -
lim 2*x^2*y^2/(x^2+y^2)= 0 -1 / lim (x^2+y^2)/2*x^2*y^2=
=-1 / lim (1/(2*x^2)+1/(2*y^2))=-1/infinity = 0;

I hope this is correct...

By the way...HallsofIvy...
I don't think your "notation" is correct...
Because if you say x=r*cost and y=r*sint you practically say
x=k*y, which is not correct...x could be equal to y^2...
(because x->0 and y->0 means that r->0, because cost and sint
can't -> 0 in the same time)
See ya...

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Homework Helper
By the way...HallsofIvy...
I don't think your "notation" is correct...
Because if you say x=r*cost and y=r*sint you practically say
x=k*y, which is not correct...x could be equal to y^2...
(because x->0 and y->0 means that r->0, because cost and sint
can't -> 0 in the same time)

I SAID that was converting to polar coordinates. The point with coordinates r and theta in polar coordinates has x= r cos(t) and
y= r sin(t) in cartesian coordinates. Believe it or not I am completely aware that as (x,y)-> (0,0), r-> 0! That was the whole point! Since r measures the distance from (0,0) to the point, the two variables x and y going to 0 reduces to the single variable r going to 0.

bogdan
(x,y) is a point ?
Not a pair of variables ?
My mistake then...
Sorry...
But there's no real need to consider them coord of a point...