Question

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{x^2 + y^5}{x^2 + y^4} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Then, which of the following is/are TRUE?

• A. The iterated limits \( \lim_{x \to 0} \left( \lim_{y \to 0} f(x, y) \right) \) and \( \lim_{y \to 0} \left( \lim_{x \to 0} f(x, y) \right) \) exist.
• B. Exactly one of the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exists at \( (0, 0) \).
• C. Both the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist at \( (0, 0) \).
• D. \( f \) is NOT differentiable at \( (0, 0) \).

Hint:

When evaluating iterated limits, check the limits one direction at a time. For multivariable functions, iterated limits may exist even if the overall limit does not.

Answer 1

The Correct Option is A

Step 1: Check the limit as \( (x, y) \to (0, 0) \).
The limit \( \lim_{(x,y) \to (0,0)} f(x,y) \) does not exist because the function behaves differently along different paths to the origin. However, we are asked to evaluate the iterated limits. 
Step 2: Compute the iterated limits.
We compute \( \lim_{x \to 0} \left( \lim_{y \to 0} f(x, y) \right) \) and \( \lim_{y \to 0} \left( \lim_{x \to 0} f(x, y) \right) \). - For \( \lim_{y \to 0} f(x, y) \) when \( x \neq 0 \), we get: \[ f(x, 0) = \frac{x^2 + 0^5}{x^2 + 0^4} = 1. \] Thus: \[ \lim_{y \to 0} f(x, y) = 1 \quad \text{for } x \neq 0. \] Now, compute \( \lim_{x \to 0} 1 = 1 \). Thus: \[ \lim_{x \to 0} \left( \lim_{y \to 0} f(x, y) \right) = 1. \] - For \( \lim_{x \to 0} f(x, y) \) when \( y \neq 0 \), we get: \[ f(0, y) = \frac{0^2 + y^5}{0^2 + y^4} = y. \] Thus: \[ \lim_{x \to 0} f(x, y) = y \quad \text{for } y \neq 0. \] Now, compute \( \lim_{y \to 0} y = 0 \). Thus: \[ \lim_{y \to 0} \left( \lim_{x \to 0} f(x, y) \right) = 0. \] Step 3: Conclusion.
The iterated limits exist, as both limit expressions give finite values. 
Final Answer: \[ \boxed{\text{The iterated limits } \lim_{x \to 0} \left( \lim_{y \to 0} f(x, y) \right) \text{ and } \lim_{y \to 0} \left( \lim_{x \to 0} f(x, y) \right) \text{ exist.}} \]