Question

Evaluate the Given limit: $\lim_{x\rightarrow 0}$ $\frac{sin\,ax}{sin\,bx}$ ,a,b ≠0

Answer 1

At x = 0, the value of the given function takes the form 0/0.
$\lim_{x\rightarrow 0}$ $\frac{sin\,ax}{sin\,bx}$ = $\lim_{x\rightarrow 0}$ ($\frac{sin\,ax}{ax}$ ) $\times$ ax / ($\frac{sin\,bx}{bx}$) $\times$ bx
$\frac{\frac{a}{b}\frac{sin\,ax}{ax}\lim_{ax\rightarrow 0}}{\lim_{bx\rightarrow 0}\frac{sin\,bx}{bx}}$[x→0 $\Rightarrow$ax → 0 and x→0 ⇒ bx → 0 ]
=($\frac{a}{b}$) $\times$ $\frac{1}{1}$ [$\lim_{y\rightarrow 0}$ $\frac{sin\,y}{y}$ = 1]
= $\frac{a}{b}$