Question

The limit $ \lim_{x \to 0} \left( \frac{\tan x - x}{x} \right) \cdot \left( \sin \frac{1}{x} \right) $ is equal to

• A. 0
• B. 1
• C. A real number other than 0 and 1
• D. None of these

Hint:

When evaluating limits involving oscillating functions, check if the other terms tend to 0, which will cause the overall limit to be 0.

Answer 1

The Correct Option is A

Step 1: Analyze the First Term
First, simplify the term \( \frac{\tan x - x}{x} \).
For small \( x \), we know: \[ \tan x \approx x + \frac{x^3}{3} \] Thus: \[ \tan x - x \approx \frac{x^3}{3} \] Now divide by \( x \): \[ \frac{\tan x - x}{x} \approx \frac{x^2}{3} \] As \( x \to 0 \), this term approaches 0.
Step 2: Analyze the Second Term
The term \( \sin \frac{1}{x} \) oscillates between -1 and 1 for all \( x \), but does not affect the limit since it is bounded.
Step 3: Conclusion
Since the first term approaches 0, the whole product approaches 0.
Hence, the limit is 0.