Proving Limits Do Not Exist Using Two Path Test

[mharten1]

Homework Statement

Use the two path test to prove that the following limits do not exist.

Homework Equations

$$\lim_{(x,y)\rightarrow{(0,0)}}\frac{4xy}{3x^2+y^2}$$

The Attempt at a Solution

The book that I am using introduces the Two Path Test theoretically but does not show an example of how to do it, so I am a bit lost.

Would I set x = y, and x = -y? In some of the more basic problems I was able to set x = 0 and y = 0, and find the limits would differ, proving that there was no limit. But in this case, that's obviously not possible.

 

[Dick]

Those two paths look like a good choice to me. Try them out. What's the limit along each path?

 

[mharten1]

Those two paths look like a good choice to me. Try them out. What's the limit along each path?

I'm getting 1 and -1, thus the limit does not exist. A question that I have that the book does not address: how do I choose the paths? Do you just try what you think will work until you find something, or is there a specific method of choosing?

 

[Dick]

I'm getting 1 and -1, thus the limit does not exist. A question that I have that the book does not address: how do I choose the paths? Do you just try what you think will work until you find something, or is there a specific method of choosing?

There's no formula for picking the paths. Just try some until you get a feeling for what's going on. Other easy ones to try are x=0 and y=0.