Question

Compute:$\underset{x\to 0}{lim}\left(\frac{\sin ax}{\sin bx}\right)$$b\ne 0,a\ne b$

Answer 1

We have,

$\underset{x\to 0}{lim}\left(\frac{\sin ax}{\sin bx}\right)$

$\underset{x\to 0}{lim}\left(\frac{a\left(\frac{\sin ax}{ax}\right)}{b\left(\frac{\sin bx}{bx}\right)}\right)$

$=\frac{a}{b}\left(\frac{\underset{x\to 0}{lim}\left(\frac{\sin ax}{ax}\right)}{\underset{x\to 0}{lim}\left(\frac{\sin bx}{bx}\right)}\right)$

We know that

$\underset{x\to 0}{lim}\left(\frac{\sin x}{x}\right)=1$

Therefore,

$=\frac{a}{b}\times \frac{1}{1}$

$=\frac{a}{b}$

Hence, the value is $1$.