Evaluate the following limits in Exercise: _ x 0 ax bx - Vidyadip

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\text{bx}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\text{bx}}$At x = 0, the value of the given function takes the form $\frac{0}{0}$.
$\text{Now},\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\text{bx}}=\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\text{ax}}\times\frac{\text{ax}}{\text{bx}}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{\sin{\text{ax}}}{\text{ax}}\Big)\times\Big(\frac{\text{a}}{\text{b}}\Big)$
$=\frac{\text{a}}{\text{b}}\lim\limits_{\text{ax}\rightarrow0}\Big(\frac{\sin{\text{ax}}}{\text{ax}}\Big)$ $[\text{x}\rightarrow0\Rightarrow\text{ax}\rightarrow0]$
$=\frac{\text{a}}{\text{b}}\times1$ $\Big[\lim\limits_{\text{y}\rightarrow0}\frac{\sin{\text{y}}}{\text{y}}=1\Big]$
$=\frac{\text{a}}{\text{b}}$

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