# Is this proof of limit existence correct?

So I have to prove the existence of the following: $\lim \limits_{(x,y) \to (0,0)} \frac{-e^{xy} + 1}{xy}$

First I attempt to find $\lim \limits_{(x,y) \to (0,0)} \frac{1}{xy}$, so should this not exist I can assert that the limit does not exist.

Looking at the plot of this function I can see that closing on to the origin the path $y = x$ approaches $\infty$ , and the path $y = -x$ approaches $-\infty$, so I think it's safe to assume that this limit does not exist.

To prove it I write the following:

$\lim \limits_{(x,y) \to (x,x)} \frac{1}{xy}=$ $\lim \limits_{(x,y) \to (x,x)} \frac{1}{x^2}=\infty$

$\lim \limits_{(x,y) \to (x,-x)} \frac{1}{xy}=$ $\lim \limits_{(x,y) \to (x,-x)} \frac{1}{-x^2}=-\infty$

Is this correct? Most methods usually replace $(x,y)$ with $(x, 0)$, $(x,mx)$, etc. so I had doubts as to whether this would be a valid answer.