Question

Prove that the following limit does not exist:

lim(x,y)->(1,1) (xy^(2)-1)/(y-1)

Answer 1

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

To show that the limit does not exist take two different pairs of sequences with limit 1

1, and show that the limit in both cases is not equal.

if we set x = y we get

$\lim_{y\to 1} \frac{y^3-1}{y-1}=\lim_{y\to 1}(y^2+y+1)=3$

if we set x = 1 we get

$\lim_{y\to 1} \frac{y^2-1}{y-1}=\lim_{y\to 1}(y+1)=2$

So the limit does not exist.

Answer: the statement that the limit does not exist is true