Limits: Behavior Of $x^2y/(x^4+y^2)$ Near The Origin

Difficult

For every point $(x,y)\ne (0,0)$ in the punctured Cartesian plane, let:

$$ f(x,y) = \cfrac{x^2y}{x^4+y^2}.$$

Select each of the correct statements (and only the correct statements) about the function $f$ from the options below.

A

The function $f$ has a removable discontinuity at the origin $(0,0)$.

B

Near the origin, the function $f$ has the following limiting be havior:

$$\lim_{(x,y)\to(0,0)} f(x,y)=0.$$

C

If we define $f(0,0)=0$, then the function $f$ becomes continuous at every point of the Cartesian plane, including the origin $(0,0)$.

D

The values of the function $f$ approach $0$ in the limit as $(x,y)$ approaches the origin $(0,0)$ along any straight line.

E

Given any real number $L$ such that $-1< L< 1$, there exists a sequence of points $(x,y)$ in the domain of $f$ such that $f(x,y)$ approaches $L$ in the limit as $(x,y)$ approaches the origin $(0,0)$.

F

It is not possible to define $f(0,0)$ so as to make the function $f$ continuous at the origin $(0,0)$.