If $f(x,y)=(x+y)\cdot \frac{y+(x+y)^2}{y-(x+y)^2}$ show that $\displaystyle \lim_{(x,y)\to(0,0)} f(x,y)$ does not exist.
Since both the iterated(repeated) limits exist and are 0, if the double limit exists it must be equal to 0. Hence I have to prove the double limit can't be 0.
I plotted $f(x,y)=0.5$ on Desmos and saw a portion of the curve is arbitrarily close to the origin but not continuous there. So no matter how small $\delta>0$ you choose, $f(x',y')=0.5>\epsilon$ for some $|x'|<\delta,|y'|<\delta$. Hence the double limit doesn't exist.
But since the question is from Analysis course, I must solve it analytically. Can anyone define $x=\phi(y)$ or $y=\psi(x)$ s.t. their limits are different for different constants used ! Although you can solve it with different approaches also.
Best Answer
First substitute $t=x+y$ to get $$\lim_{(x,y)\to(0,0)}f(x,y) = \lim_{(t,y)\to(0,0)}t\cdot\frac{y+t^2}{y-t^2}$$ Now substituting $y=kt^3+t^2$ gives the limit $\frac2k$, for any constant $k$. So limit doesn't exist.