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]
= \(\frac{a}{b}\)