PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Functions of Several Variables
Let f : ℝ2 → R be defined as follows:
f(x, y) = \(\left\{\begin{array}{cc} \rm \frac{x^4 y^3}{x^6+y^6} & \text { if } \rm (x, y) \neq(0,0) \\ 0 & \text { if }\rm (x, y)=(0,0) . \end{array}\right.\)
Then
1
\(\rm \displaystyle\lim _{t \rightarrow 0} \frac{f(t, t)-f(0,0)}{t} \) exists and equals \(\frac{1}{2}\)
2
\(\rm \left.\frac{\partial f}{\partial x}\right|_{(0,0)}\) exists and equals 0
3
\(\rm \left.\frac{\partial f}{\partial y}\right|_{(0,0)} \) exists and equals 0
4
\(\rm \displaystyle\lim _{t \rightarrow 0} \frac{f(t, 2 t)-f(0,0)}{t}\) exists and equals \(\frac{1}{3}\)