Question

$\underset{x\to 0}{lim}\frac{{\log }_{{\sin }^{2}x}\cos x}{{\log }_{{\sin }^2\frac{x}{2}}\cos \frac{x}{2}}$ has the value equal to

• A. 1
• B. 2
• C. 4
• D. None of these

Answer 1

Answer: (C)

4

$\underset{x\to 0}{lim}\frac{\log \cos x}{\log \cos \frac{x}{2}}\cdot \frac{\log \left|\sin \frac{x}{2}\right|}{\log \left|\sin x\right|}$
$=\underset{x\to 0}{lim}\frac{\log \cos x}{\log \cos \frac{x}{2}}\times \underset{x\to 0}{lim}\frac{\log \left|\sin \frac{x}{2}\right|}{\log \left|\sin x\right|}$
$=\underset{x\to 0}{lim}\frac{\tan x}{\tan \frac{x}{2}}\times \underset{x\to 0}{lim}\frac{\cot \frac{x}{2}}{\cot x}=\underset{x\to 0}{lim}\frac{{\tan }^{2}x}{{\tan }^2\frac{x}{2}}=4$