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

Concept: Use identity: \[ \sin 3x = 3\sin x - 4\sin^3 x \]

Step 1:Substitute.
\[ \frac{\sin 3x - \sin x}{\sin x} = \frac{(3\sin x - 4\sin^3 x) - \sin x}{\sin x} \] \[ = \frac{2\sin x - 4\sin^3 x}{\sin x} \]

Step 2:Simplify.
\[ = 2 - 4\sin^2 x \]

Step 3:Apply limit.
\[ \lim_{x\to 0} (2 - 4\sin^2 x) = 2 - 0 = 2 \]