Link to solution. It is problem number 3.
$\lim_{(x,y) \to (0,0)}\frac{y^2 \sin^2 x}{x^4 + y^4}$
Solution: When $x$ is small [close to $0$], $\sin x$ has essentially the same growth as $x$. So the numerator is like $x^2y^2$, has the same degree as denominator. In this scenario, it's likely that the limit does not exist. To test this, we will try $y = mx$, where $m$ is just any real number. Then, we get $\frac{y^2 \sin^2 x}{x^4 + y^4} = \frac{m^2x^2 \sin^2 x}{(1 + m^4)x^4} = \frac{m^2 \sin^2 x}{(1 + m^4)x^2}$ Take the limit as $x$ goes to $0$, we will get $\frac{m^2}{1 + m^4}$ , which in particular means if we take $m = 2, m = 3$, the answers will not agree, and hence the limit does not exist.
I do not understand the last paragraph.