Evaluate the Given limit:limits_{x \to 0}{sinax}{bx}

\( \lim\limits_{x \to 0} \dfrac{sinax}{bx}\)

At x = 0, the value of the given function takes the form \( \dfrac{0}{0}\)

Now,

\( \lim\limits_{x \to 0} \dfrac{sin\;ax}{bx}\)

= \( \lim\limits_{x \to 0} \dfrac{sin\;ax}{ax}\times \dfrac{ax}{bx}\)

= \( \lim\limits_{x \to 0} \dfrac{sin\;ax}{ax}\times \dfrac{a}{b}\)

= \(\dfrac{a}{b} \lim\limits_{ax \to 0} \dfrac{sin\;ax}{ax} \)

= \( \dfrac{a}{b}\times 1\)

= \( \dfrac{a}{b}\)