Question

Find c

$\underset{x\to 0}{lim}\frac{\log \tan (\frac{\pi }{4}+ax)}{\sin bx}=c\frac{a}{b}.$

Answer 1

$\underset{x\to 0}{Lt}\frac{\log \tan (\frac{\pi }{4}+ax)}{\sin bx}$
$=\underset{x\to 0}{Lt}\frac{\log (1+\tan ax)-\log (1-\tan ax)}{\sin bx}$
$=\underset{x\to 0}{Lt}\frac{2[\tan ax+\frac{1}{3}{\tan }^{3}ax\cdots ]}{\sin bx}$
$=\underset{x\to 0}{Lt}\frac{2\tan ax}{\sin bx}[1+\frac{1}{3}{\tan }^{2}ax+....]=2\frac{a}{b}.$