Gate 2016

Question

Answer 1

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