Q46Multiple correct3 Marks24 Dec 2023

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function given by: $$f(x, y) = \begin{cases} \frac{(x+y)^2}{x^2+y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}$$ Choose the correct option(s) from the following:

A

lim_{(x,y)->(0,0)} f(x,y) exists along the positive x-axis.

B

f(x,y) approaches 0 when $(x,y)$ approaches $(0,0)$ along the positive y-axis.

C

lim_{(x,y)->(0,0)} f(x,y) does not exist.

D

f_y(0,0) = 1.

E

The directional derivative of f at $(0,0)$ in the direction of positive x-axis does not exist.

F

The directional derivative of f at $(0,0)$ in the direction of positive y-axis exists.