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

The Correct Option is A, B, C

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.