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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?

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

1 Answer 1

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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.)

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