Question Number 11395 by Joel576 last updated on 23/Mar/17

Prove that those functions below don′t have limit  a) lim_((x,y)→(0,0))   ((xy)/(x^2  + y^2 ))    b)  lim_((x,y)→(0,0))   ((xy + y^3 )/(x^2  + y^2 ))

Commented byprakash jain last updated on 23/Mar/17

a)  x=rcos θ  y=rsin θ    ((xy)/(x^2  + y^2 ))=((r^2 cos θsin θ)/r^2 )= ((sin 2θ)/2)  lim_((x,y)→(0,0))   ((xy)/(x^2  + y^2 ))=lim_(r→0) ((sin 2θ)/2)=((sin 2θ)/2)  The limit value depends on the angle of  the line chosen to approach (0,0) so the  limit does not exist.