<p>The value of limx→0
ln(sin 3x)
ln(sin x) is:</p>
Step-by-Step Solution
Key Concept: This is a \infty/\inftyform. Logarithmic limits of similar functions often reduce to the ratio of their
coefficients.
<p><strong>1</strong>: As x \to 0, sin 3x \to 0+ and sin x \to 0+, so ln(0+) \to -\infty.</p><p><strong>2</strong>: Apply L’Hopital: \lim_{x \to 0</p>} 3 cos 3x/ sin 3x cos x/ sin x = 3 \lim_{x \to 0} cot 3x cot x .<p><strong>3</strong>: Rewrite as: 3 \lim_{x \to 0} tan x</p> tan 3x.<p><strong>4</strong>: Using lim\theta\to 0 tan \theta</p> \theta = 1, we have 3 x 3x = 1.
Correct Answer: 2