How to Show limit does not exist

Question

Show that the limit does not exist: lim _{(x,y)→(0,0)} sin (xy) /(x ² + y)

Answer 1

big correction,

if the problem was L= lim x,y ->0 sin(xy)/(x²+y²)

then can show L does not exist

when y=x, L=1/2, when y=y L=0 therefore no single L

But for L= lim x,y ->0 sin(xy)/(x²+y)

L appears to be one number, L=0

Answer 2

Hi Nina L.,

Without even applying L'Hopital's Rule, I would bet you that the value you get depends on which direction x and y variously approach zero from? But even more major here, is the sin(x)/x^2 consideration: that blows up to infinity, doesn't it (since sin(x)/x approaches 1). Infinity is not a limit, last time I checked?

-- Cheers, --Mr. d.