Question

Let \( f: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R} \) be a function defined by

\[ f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} + x \sin \left( \frac{1}{x^2 + y^2} \right). \]

Consider the following three statements:

S1: \( \lim_{x \to 0,\, y \to 0} f(x, y) \) exists.

S2: \( \lim_{y \to 0} \lim_{x \to 0} f(x, y) \) exists.

S3: \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists.

Then, which of the following is/are correct?

• A. S2 and S3 are TRUE and S1 is FALSE
• B. S1 and S2 are TRUE and S3 is FALSE
• C. S1 and S3 are TRUE and S2 is FALSE
• D. S1, S2 and S3 are all TRUE

Hint:

When checking limits in multiple variables, always test the behavior along different paths to see if the limit is path-dependent. If the limit depends on the direction, the two-variable limit does not exist.

Answer 1

The Correct Option is B

To determine which of the statements S1, S2, and S3 are true, we will analyze the behavior of the function \( f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} + x \sin \left( \frac{1}{x^2 + y^2} \right) \) as \( (x, y) \to (0, 0) \).

1. To examine S1: Analyze \( \lim_{x \to 0, y \to 0} f(x, y) \):
   1. Using polar coordinates, set \( x = r \cos \theta \) and \( y = r \sin \theta \), then \( x^2 + y^2 = r^2 \).
   2. Rewrite \( f \) as: f(r \cos \theta, r \sin \theta) = \frac{(r \cos \theta)^2 - (r \sin \theta)^2}{r^2} + r \cos \theta \sin \left(\frac{1}{r^2}\right)
   3. This simplifies to: f(r \cos \theta, r \sin \theta) = \cos(2\theta) + r \cos \theta \sin \left(\frac{1}{r^2}\right)
   4. Observe that the term \(\cos(2\theta)\) depends on \(\theta\) meaning it is path-dependent. Hence, \( \lim_{x \to 0, y \to 0} f(x, y) \) does not exist uniformly because it depends on the path \(\theta\) being taken to reach the origin.
2. For S2: Examine \(\lim_{y \to 0} \lim_{x \to 0} f(x, y) \):
   1. First consider \(\lim_{x \to 0} f(x, 0) = \frac{x^2}{x^2} + x\sin \left( \frac{1}{x^2} \right) = 1 + x\sin \left( \frac{1}{x^2} \right)\).
   2. Since \(-1 \leq \sin \left( \frac{1}{x^2} \right) \leq 1\), the term \(x \sin \left( \frac{1}{x^2} \right)\) oscillates and approaches 0 as \(x \to 0\).
   3. Thus, \(\lim_{x \to 0} f(x, 0) = 1\).
   4. Now, \(\lim_{y \to 0} 1 = 1\) indicates that \(\lim_{y \to 0} \lim_{x \to 0} f(x, y)\) exists and equals 1.
3. For S3: Examine \(\lim_{(x, y) \to (0, 0)} f(x, y) \):
   1. The analysis under S1 showed path dependence due to the term \(\cos(2\theta)\). Therefore, \(\lim_{(x, y) \to (0, 0)} f(x, y)\) does not exist uniformly.

In conclusion, the correct observations are: S1 is TRUE because the limit does not exist uniformly due to path dependency, S2 is TRUE because the iterated limits exist, and S3 is FALSE because the limit does not exist uniformly. Therefore, the correct option is: S1 and S2 are TRUE and S3 is FALSE.