Evaluate the Given limit:

Question:

Evaluate the Given limit: $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$

Solution:

$\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$

At $x=0$, the value of the given function takes the form $\frac{0}{0}$.

Now, $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}=\lim _{x \rightarrow 0} \frac{\sin a x}{a x} \times \frac{a x}{b x}$

$=\lim _{x \rightarrow 0}\left(\frac{\sin a x}{a x}\right) \times\left(\frac{a}{b}\right)$

$=\frac{a}{b} \lim _{x \rightarrow 0}\left(\frac{\sin a x}{a x}\right)$ $[x \rightarrow 0 \Rightarrow a x \rightarrow 0]$

$=\frac{a}{b} \times 1$ $\left[\lim _{y \rightarrow 0} \frac{\sin y}{y}=1\right]$

$=\frac{a}{b}$

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