How to Prove the Limit of (x^4+y^4)/(x^2+y^2) is 0 at (0,0)?

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And, of course, the original questioner wanted something that would work in general. In general, you cannot assume that a function of two variables can be written as a function of a single variable. In general, you need to find a path along which the limit is easy to calculate. Here, converting to polar coordinates was the easiest method.In summary, the conversation discusses ways to prove that the limit of the function (x^4+y^4)/(x^2+y^2) as (x,y) approaches (0,0) is equal to 0. Various methods are suggested, including using the epsilon-delta definition of a limit, trying a clever substitution, and using L'Hospital's rule. The
  • #1
Paradox
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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  • #2
Apply the epsilon-delta definition of a limit, or try a clever substitution.

Hurkyl
 
  • #3
Thank you for the quick response. I'll go and try it out now.
 
  • #4
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.
 
  • #5
Oops, wrong forum, wrong symbols. Well, the math is still correct.
 
  • #6
Would using L'Hospital's rule (successive differentiations of numerator and denominator) be cheating?
 
  • #7
Since this is a function of two variables, HOW, exactly, would you apply L'Hospital's rule?
 
  • #8
Thanks HallsOfIvy. That was much easier to work with
 
  • #9
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.
 
  • #10
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.
 
  • #11
ObsessiveMathsFreak
Not that when they are zero the value of the function is NOT zero.
Is that value then undefined?
 
  • #12
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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  • #13
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.
 
  • #14
(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...
 
  • #15
If you are willing to ASSUME that you can you can
treat 1/(1/x^2+ 1/y^2) as x and y going to 0 as
"1/(infinity+infinity)" and declare that "1/infinity" is 0, then there is no real need to be precise at all. Yes, your method does work (and is very clever) in this example but I suspect that most mathematics professors would want you to show a little more understanding of what you are doing.
 

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function is approaching, rather than the actual value at that point.

2. Why are limits important?

Limits are important because they allow us to understand the behavior of functions and make predictions about their values. They also play a crucial role in calculus, as they are used to define derivatives and integrals.

3. How do you find the limit of a function?

To find the limit of a function, you can either use algebraic techniques such as factoring and simplifying, or you can use graphical methods by plotting the function and observing its behavior as the input approaches the specified value. You can also use calculus techniques such as L'Hôpital's rule.

4. What are the different types of limits?

The three main types of limits are one-sided limits, two-sided limits, and infinite limits. One-sided limits describe the behavior of a function as the input approaches a value from either the left or the right, while two-sided limits describe the behavior as the input approaches the value from both sides. Infinite limits occur when the output of a function approaches positive or negative infinity as the input approaches a certain value.

5. How are limits used in real-life applications?

Limits are used in various real-life applications, such as physics, engineering, and economics. For example, in physics, limits are used to study the behavior of particles as they approach the speed of light. In engineering, limits are used to analyze the stability and performance of systems. In economics, limits are used to model and predict economic growth and trends.

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