Question

Choose the most appropriate option. \[ \lim_{x \to 1} \sin(x - 1) \cdot \tan\left( \frac{\pi}{x} \right) \]

• A. 0
• B. \( \frac{1}{\pi} \)
• C. \( \frac{2}{\pi} \)
• D. \( -\frac{3}{\pi} \)

Hint:

When a term tends to 0 and the other is finite, the product tends to 0.

Answer 1

The Correct Option is A

As \( x \to 1 \), the term \( \sin(x - 1) \to 0 \) and \( \tan\left( \frac{\pi}{x} \right) \) is finite, so the product of these two terms tends to 0: \[ \lim_{x \to 1} \sin(x - 1) \cdot \tan\left( \frac{\pi}{x} \right) = 0 \]