Limit as (x,y) -> (0,0) ln(x^2+y^2)

1. May 15, 2012

theBEAST

I know the answer is -infinity but I am not sure how to prove this. Would I just set x=0 and say that lim y-> ln(y^2) = -inf and do the same with y=0? But this would not prove it for all cases... Do I have to use the definition of the limit?

2. May 16, 2012

Fightfish

The form (x^2+y^2) strongly suggests a parameterisation, yes?

3. May 16, 2012

theBEAST

What does that mean xD, I just started learning limits with two variables today and I found this question online and found it interesting. Something tells me my current knowledge is insufficient to solve this problem...

4. May 16, 2012

Fightfish

Ah, so you get to learn a new technique today:).

Let's use polar coordinates:
$$x = r cos \theta, y = r sin \theta$$
Then our limit becomes
$$\lim_{r \to 0} ln (r^2) = 2 \lim_{r \to 0} ln (r)$$​
You realise here that we have managed to remove the dependence on the direction of approach - ie our answer is independent of $\theta$.

5. May 16, 2012

theBEAST

Wow this is a pretty cool way to solve limits, thanks!

I have one more question:
Does the limit as (x,y,z) -> (0,0,0) (xy+yz^2+xz^2)/(x^2+y^2+z^4) exist?

What I did was I set x,y = 0 and solved the limit as z -> 0 which gave me zero.

Next I set y=z^2 and x=z^2 and found the limit as z->0 which gave me 1. Therefore the limit does not exist.

Is this a correct method? I am not sure if it is legal to independently set y and z equal to z^2.

PS: the textbook did it differently they set y=x and z=0 and let the limit x -> 0 but I think there should be multiple ways to do this.

6. May 16, 2012

Fightfish

Yup. You can choose to approach the limit from any direction that you like. In your second case, you were approaching it from a parabolic path; that's fine.
Yes. A multivariable limit exists at a point only if the limit gives the same value no matter which path you take as you approach the point. To prove that a multivariable limit does not exist, it thus suffices to either show that 1) any two different paths give different limits or 2) there is a path for which the limit does not exist.

The tricky part in multivariable limits is to prove that a limit does exist, because you must show that the limit attains the same value for all possible paths of approach.