Question

Let \(f(x, y) = \begin{cases} \frac{xy}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\), then

• A. $\lim_{(x, y) \to (0, 0)} f(x, y)$ does not exist.
• B. $f(x, y)$ is differentiable at $(0, 0)$.
• C. $f(x, y)$ is continuous but not differentiable at $(0, 0)$.
• D. $\lim_{(x, y) \to (0, 0)} f(x, y)$ exists, but $f(x, y)$ is not continuous at $(0, 0)$.

Hint:

For functions of the form $f(x,y) = \frac{x^a y^b}{(x^2+y^2)^c}$:
- Continuous at $(0,0)$ if $a+b > 2c$.
- Differentiable at $(0,0)$ if $a+b > 2c + 1$.
Here $a=1, b=1, c=1/2$, so $a+b = 2 > 2(1/2) = 1$ (Continuous!), but $a+b = 2 \ngtr 1+1 = 2$ (Not Differentiable!).

Answer 1

The Correct Option is C