Question

Let \( f(x) = \frac{x + |x|(1 + x)}{x} \sin \left( \frac{1}{x} \right) \), for \( x \neq 0 \). Write \( L = \lim_{x \to 0^-} f(x) \) and \( R = \lim_{x \to 0^+} f(x) \). Then which one of the following is TRUE?

• A. \( L \) does not exist but \( R \) exists
• B. \( L \) exists but \( R \) does not exist
• C. Both \( L \) and \( R \) exist
• D. Neither \( L \) nor \( R \) exists

Hint:

Check the behavior of absolute value functions carefully for left and right limits when dealing with piecewise functions.

Answer 1

The Correct Option is B

Step 1: Analyzing the limits.
First, consider the left-hand limit \( L \) as \( x \to 0^- \). We have the term \( |x| \), which behaves differently for negative \( x \), leading to the non-existence of the limit. For the right-hand limit \( R \), we have the term \( \sin \left( \frac{1}{x} \right) \), which oscillates but remains bounded as \( x \to 0^+ \).
Step 2: Conclusion.
Since the left-hand limit \( L \) exists, but the right-hand limit \( R \) does not exist, the correct answer is (A).