Limit of a Function of Two Variables

In summary, the problem involves evaluating the limit of the function f(x,y) = \frac{y^3}{x^2+y^2} as x and y approach the origin. Using the multiple-path method and a polar substitution, the limits are found to be y and 0, respectively. The squeeze theorem, also known as the sandwich theorem or pinching theorem, can be applied by using the fact that x^2 + y^2 \geq y^2.
  • #1
Mindstein
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Homework Statement


Evaluate the limit of the function f(x,y) = [tex]\frac{y^3}{x^2+y^2}[/tex]


Homework Equations





The Attempt at a Solution


Well, I approached this problem using the multiple-path method and found the following:

[tex]\stackrel{lim}{x\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y
[tex]\stackrel{lim}{y\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = 0

and am having trouble interpreting these results. I tried doing a polar substitution and found that:

[tex]\stackrel{lim}{r\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y*sin2(theta)

My calculus book is very short on the topic, I am pretty much left in the dark. Please bring me into the light. haha.
 
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  • #2
Are you trying determine the limit of the function at the origin? Try using the fact that [itex]x^2 + y^2 \geq y^2[/itex] and apply the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken]
 
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  • #3
snipez90 said:
Are you trying determine the limit of the function at the origin? Try using the fact that [itex]x^2 + y^2 \geq y^2[/itex] and apply the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken]

Yes, I am trying to determine the limit as x,y approaches the origin. The problem is that I have no clue how to apply the squeeze theorem. I can't find any good sites on it.
 
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  • #4
google squeeze theorem - it's also called the sandwich theorem, or the pinching theorem
 
  • #5
Ahhh okay, so what I would want to say is that [tex]y^2[/tex] [tex]\leq[/tex][tex]\frac{y^3}{x^2+y^2}[/tex][tex]\leq[/tex]x2+y2 ?
 

What is the definition of a limit of a function of two variables?

The limit of a function of two variables is the value that a function approaches as the input values approach a given point in the domain. In other words, it is the value that the function "approaches" or gets closer to as the input values get closer to a specific point.

How is the limit of a function of two variables calculated?

The limit of a function of two variables can be calculated by approaching the given point along different paths (or directions) and checking if the function values approach the same value. If they do, then the limit exists and is equal to that common value. If the values approach different values, then the limit does not exist.

What is the difference between a limit and a value of a function?

The limit of a function of two variables is the value that the function "approaches" as the input values get closer to a specific point. The value of a function, on the other hand, is the actual output value of the function at a given point. While the limit can be calculated by approaching a point, the value of a function can only be calculated by plugging in the input values.

What is a one-sided limit of a function of two variables?

A one-sided limit of a function of two variables is the limit of the function as the input values approach a given point from only one side. This means that the input values are either approaching the point from the left or from the right. One-sided limits are useful in cases where the function may behave differently on each side of the point.

What types of functions can have a limit of a function of two variables?

Any function that takes in two variables as inputs can have a limit of a function of two variables. This includes polynomial functions, rational functions, exponential functions, trigonometric functions, and more. However, not all functions will have a limit at every point in their domain. Some may have limits at certain points and not others, while some may not have a limit at all.

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