limits multivariable calculus. where am i wrong with my attempt?

Question

P : $\lim_{(x,y) \to (0,0)} f(x,y)$ where
$$f(x,y) = y\sin\frac1x + \frac{xy}{x^{2} + y^{2}}$$

Text book says Limit doesnot exist . So where i am wrong with my proof below ?

EDITED ATTEMPT :

Or we can write $2 \delta < \varepsilon$, which shows limit exists. Is there something I missed?

Answer 1

$y>y^2$ for $y\in(0,1)$, which is the source of mistake.

Putting $y=cx$ gives $$ cx\sin(1/x)+\frac{c}{1+c^2} $$ and the limit in 0 certainly doesn't exist, because the first term tends to 0.

Answer 2

If $|f| \leq g$ and $g \to 1/2$, you can't conclude that $f \to 1/2$! You must have $g \to 0$ to conclude.

The limit doesn't exist: pick $y=x$ and compute $\lim_{x \to 0} f(x,x)$. Then pick $y=0$ and compute $\lim_{x \to 0} f(x,0)$

Answer 3

The mistake is here :

for $x$ and $y$ tending to $0$, then $\arrowvert{ \frac{y}{x^2+y^2}}\arrowvert$ tends to infinity

$\arrowvert{ \frac{xy}{x^2+y^2}}\arrowvert$ is undeterminated