I am new to using two-path test and my textbook only discusses it without showing any examples. I attempted to do this question below but I am not sure if I am correct. The question says to show the limit doesn't exist as $(x,y) \to (0,0)$:
$$f(x,y)=\frac{xy}{|xy|}.$$
First I set $y=0$ and let $x \to 0$ and got the limit to be undefined
Second I set $x=0$ and let $y \to 0$ and got the limit to be undefined
Is this how you do this test? Since limits are undefined they don't exist at this point $(0,0)$.
Best Answer
In your argument, you are choosing points which are not in the domain of the function. This is illegal. What you want to do is approach $(0,0)$ via points that lie in the domain. For instance, if you choose a line $y=mx$, then along this line, $$ f(x,y) = \frac{m}{|m|} $$ for any point $x\neq 0$. Thus, the limit value will depend on the sign of $m$!
So consider the line $y=x$, and you will end up with $$ f(x,y) = 1 $$ for all points on this line. So when you take the limit, you get 1.
Now consider the line $y=-x$, and take the limit, you get $-1$.
This tells you that the limit cannot exist.
Added : If you want to play around with this idea, try looking at the limits of these functions at $(0,0)$ : $$ f(x,y) = \frac{xy}{x^2 + y^2} $$ $$ f(x,y) = \frac{xy}{x^3 + y^{3/2}} $$