As (x,y) approaches (a,b) the limit of f(x,y) is L,if the limits from all paths approaching (a,b) exists and are all equall to L
if a function is not continuous at point a then limit doesnot exist for one variable and is same for 2 variable limit also.
$lim_{(x,y)->(a,b)}f(x,y)=f(a,b)$
at two paths y=x and y=-x the value should be same then limit exist at that point.otherwise limit doesnot exist same as one variable limit
$\lim_{x->0 and y->0} \frac{xy}{x^{2}+y^{2}}$
i)y=x
then $\lim_{x->0 and y->0} \frac{xy}{x^{2}+y^{2}}$
= $x^{2}/(x^{2}+x^{2})$
=1/2.
ii)y=-x
then $\lim_{x->0 and y->0} \frac{xy}{x^{2}+y^{2}}$
=$-x^{2}/(x^{2}+x^{2})$
=-1/2.
so value is not same for two paths .
Hence limit doesnot exists at x=0 and y=0
