Question

\(\lim_{x\to0} \frac{\sin3x - \sin x}{\sin x}\) is

• A. -2
• B. 2
• C. 0
• D. None of these

Hint:

As \(x \to 0\), \(\sin x \to 0\). Use identities first, then apply limit.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We can solve this limit by splitting the fraction and applying the standard limit \( \lim_{x \to 0} \frac{\sin ax}{x} = a \).
Step 2: Detailed Explanation:
The given expression is: \[ \lim_{x \to 0} \left( \frac{\sin 3x}{\sin x} - \frac{\sin x}{\sin x} \right) \] \[ = \lim_{x \to 0} \left( \frac{\sin 3x}{\sin x} - 1 \right) \] To evaluate the first part, divide both the numerator and denominator by \( x \): \[ \lim_{x \to 0} \frac{\sin 3x}{\sin x} = \lim_{x \to 0} \frac{(\frac{\sin 3x}{x})}{(\frac{\sin x}{x})} \] Applying the standard limits: \[ = \frac{3}{1} = 3 \] Substituting this back into the original expression: \[ 3 - 1 = 2 \].
Step 3: Final Answer:
The value of the limit is 2.