Question

Let $f ∶ R^2 → R$ be a function defined as
$f(x,y) =\begin{cases}\frac{x^2y}{x^4 + Y^2} & if (x, y) \neq (0, 0)\\\ 0 & if (x, y) = (0, 0)\end{cases}$ Then, which of the following is/are CORRECT?

• A. $\lim_{(x,y) \rightarrow (0,0)} 𝑓(x, y) = 0$
• B. $f_x(0, 0) = 0$
• C. 𝑓(x, y) is not continuous at (0, 0)
• D. Both $f_x$ and $f_y$ do not exist at (0, 0)

Answer 1

Answer: (B), (C)

$f_x(0, 0) = 0$

Answer 2

The correct Options are B and C : $\lim_{(x,y) \rightarrow (0,0)} 𝑓(x, y) = 0$ AND 𝑓(x, y) is not continuous at (0, 0)