2-Path Test for Non-Existence

Find the limit or show that it does not exist:
$$\lim _{\left(x,\:y\right)\to \left(1,\:-1\right)}\left(\frac{xy+1}{x^2-y^2}\right)$$

For this question i have used 2 different paths:

Path 1: $$x=0$$, where,
$$\lim _{\left(y\right)\to \left(-1\right)}\left(\frac{1}{-y^2}\right) = -1$$

Path 2: $$y=0$$, where,
$$\lim _{\left(x\right)\to \left(1\right)}\left(\frac{1}{x^2}\right) = 1$$

This yielded 2 different limits, hence the limit does not exist. however, i have been told that i cannot use path x=0, why so? Can i use any path i want or is there a method to choose a path?

• This seems fine to me. Who said you cannot use $x =0$? Mar 14, 2019 at 18:52

Of course you can use such "path limits" only to prove that some two variables limit does not exist. Now, since the limit point is $$(1,,-1)$$, you can use the paths $$x=1$$ and $$y=-1$$. If the limits $$\lim_{y\to-1} f(1,y),\quad \lim_{x\to 1} f(x,-1)$$ are different you are done. If they are equal stronger measures will have to be taken. (The paths $$x=0$$ and $$y=0$$ have nothing to play here.)