Question

Let f : $\R^2 → \R$ be defined as follows :
$f(x,y)=\begin{cases} \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\\ 0 & \text{if } (x,y)=(0,0) \end{cases}$
Then

• A. $\lim\limits_{t \rightarrow 0}\frac{f(t,t)-f(0,0)}{t}$ exists and equals $\frac{1}{2}$
• B. $\frac{∂f}{∂x}|_{(0,0)}$ exists and equals 0
• C. $\frac{∂f}{∂y}|_{(0,0)}$ exists and equals 0
• D. $\lim\limits_{t \rightarrow 0}\frac{f(t,2t)-f(0,0)}{t}$ exists and equals $\frac{1}{3}$

Answer 1

Answer: (A), (B), (C)

$\lim\limits_{t \rightarrow 0}\frac{f(t,t)-f(0,0)}{t}$ exists and equals $\frac{1}{2}$

Answer 2

The correct option is (A) : $\lim\limits_{t \rightarrow 0}\frac{f(t,t)-f(0,0)}{t}$ exists and equals $\frac{1}{2}$, (B) : $\frac{∂f}{∂x}|_{(0,0)}$ exists and equals 0 and (C) : $\frac{∂f}{∂y}|_{(0,0)}$ exists and equals 0.