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.