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}\leqx²+y² ?