Limit of a Function of Two Variables

[Mindstein]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

$$\stackrel{lim}{x\rightarrow 0}$$ $$\frac{y^3}{x^2+y^2}$$ = y
$$\stackrel{lim}{y\rightarrow 0}$$ $$\frac{y^3}{x^2+y^2}$$ = 0

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

$$\stackrel{lim}{r\rightarrow 0}$$ $$\frac{y^3}{x^2+y^2}$$ = 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.

 

[snipez90]

Are you trying determine the limit of the function at the origin? Try using the fact that $x^2 + y^2 \geq y^2$ and apply the http://en.wikipedia.org/wiki/Squeeze_theorem"

 

[Mindstein]

Are you trying determine the limit of the function at the origin? Try using the fact that $x^2 + y^2 \geq y^2$ and apply the http://en.wikipedia.org/wiki/Squeeze_theorem"

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.

 

[statdad]

google squeeze theorem - it's also called the sandwich theorem, or the pinching theorem

 

[Mindstein]

Ahhh okay, so what I would want to say is that $$y^2$$ $$\leq$$$$\frac{y^3}{x^2+y^2}$$$$\leq$$x²+y² ?