Continuity in Calc III problem

[Telemachus]

Homework Statement

I must say if the function is continuous in the point (0,0). Which is $$\displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0)$$

The function:

$$f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix}$$

I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.

$$\displaystyle\lim_{(x,y) \to{(0,0)}}{(x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2})}$$

What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.

 

[snipez90]

Your intuition is on target. Why not try using the definition of the limit?

 

[CompuChip]

If it is not continuous, then the limit does not exist.
Clearly, if you approach (0, 0) along the line y = -x, f(x, y) tends to 1.
To show that the limit does not exist, it suffices to find one other direction for which the limit is not 1.
For example, if you can prove that
$$\lim_{x \to 0} f(x, 0) \neq 1$$
you are done.

You might find it useful that
$$\lim_{x \to \infty} \frac{\sin(x)}{x} = 0$$

 

[Telemachus]

Thank you both.

Snipez90, you mean the delta epsilon definition? I always have trouble by using it. I find it tedious, but I'd really like to improve that, cause I fill as I could never use it, and I think its necessary to make the proves. But I'm not strong on the algebraic skills that's needed to make that kind of proofs, or any kind, it's something I must learn to do really. If you know any book or something that I could use to learn some proofs techniques I'd be really thankful.