If f(x,y) = (xy-x+y)/(xy+x+y) where xy+x+y not equal to 0, then show that?

Question

lim as y --> 0 { lim as x --> 0 f(x,y)} is not equal to lim as x--> 0 { lim as y--> 0 f(x,y)}.

Do i have to solve the limits in the brackets first after which i get the solution as:
lim as y-->0 {1} not equal to lim as x--> 0 {(-1)}.
So the final answer i'm getting is 1 not equal to (-1).
Is that right way of solving this problem?

Answer 1

f(x,y) = (xy-x+y)/(xy+x+y)

lim y-->0 {lim x-->0 f(x,y)}
=lim y-->0 {lim x-->0 (xy-x+y)/(xy+x+y)}
=lim y-->0 {lim x-->0 (y(x+1)-x)/(y(x+1)+x)}
=lim y-->0 {y(0+1)-0)/(y(0+1)+0}
=lim y-->0 {y/y}
=lim y-->0 {1}
= 1

lim x-->0 {lim y-->0 f(x,y)}
=lim x-->0 {lim y-->0 (xy-x+y)/(xy+x+y)}
=lim x-->0 {lim y-->0 (x(y-1)+y)/(x(y+1)+y)}
=lim y-->0 {x(0-1)+0)/(x(0+1)+0}
=lim y-->0 {-x/x}
=lim y-->0 {-1}
= -1

Yes, lim as y-->0 {1} not equal to lim as x--> 0 {(-1)}.

Answer 2

Sounds right to me.