I'm trying to understand the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2-y^2)}{x^2-y^2}$$
The $\lim_{(x,y)\to (0,0)} f(x,y)$ is undefined. Why it is not equal to $1$?
Let's suppose $t = x^2-y^2$. Then as $(x,y)$ approaches $(0,0)$, $t$ approaches $0$, so
$$\lim_{t\to 0} \frac{\sin t}{t} = 1 $$
I can see that the path $x=y$ is undefined,does this mean that the limit does not exist ?
I though that I needed to find two different defined paths that gives two different results to disprove that the limit exists. If so this means that I can't really rely on the "t substitution" to determined the limit.
I tried to pass it into polar coordinates but It didn't help.. How can I determine a limit of a two variable function ? I can't check all the possibles paths like in one variable (left and right ).
I understood that the safest way is to use the squeeze theorem or to pass the coordinates into polar coordinates and then use the squeeze theorem.
I would appreciate any sort of help in the matter , Thanks.