Question

The value of $\underset{\theta \to 0}{lim}\frac{1-cos4\theta }{1-cos6\theta }$ is

• A. 9/4
• B. 3/4
• C. 4/9
• D. 9/3

Answer 1

The correct option is C 4/9

$\underset{x\to 0}{lim}\frac{1-\cos 4\theta }{1-\cos 6\theta }$

Put $\theta =0$ we get $\frac{0}{0}$form

So, Apply L-Hospital Rule

$=\underset{x\to 0}{lim}\frac{-[-4\sin 4\theta ]}{-[-6\sin 6\theta ]}$

$=\underset{x\to 0}{lim}\frac{4\sin 4\theta }{6\sin 6\theta }$

$=\frac{4}{6}\underset{x\to 0}{lim}\frac{\sin 4\theta }{4\theta }\cdot 4\theta \cdot \frac{6\theta }{\sin 6\theta }\cdot \frac{1}{6\theta }$

$=\frac{2}{3}\underset{x\to 0}{lim}1\cdot 4\theta \cdot 1\cdot \frac{1}{6\theta }$

$=\frac{2}{3}\cdot \frac{2}{3}$

$=\frac{4}{9}$