Evaluate the following limits in Exercise: _ x 0 ax ,a,b 0 - Vidyadip

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\sin\text{bx}},\text{a},\text{b}\neq0$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\sin\text{bx}},\text{a},\text{b}\neq0$ At x = 0, the value of the given function takes the form ${\frac{0}{0}}$. $\text{Now},$$\lim\limits_{\text{x}\rightarrow0}\frac{\sin{\text{ax}}}{\sin\text{bx}}=\lim\limits_{\text{x}\rightarrow0}\frac{\Big(\frac{\sin\text{ax}}{\text{ax}}\Big)\times\text{ax}}{\Big(\frac{\sin\text{bx}}{\text{bx}}\Big)\times\text{bx}}$$=\big(\frac{\text{a}}{\text{b}}\big)\times\frac{\lim\limits_{\text{ax}\rightarrow0}\Big(\frac{\sin\text{ax}}{\text{ax}}\Big)}{\lim\limits_{\text{bx}\rightarrow0}\Big(\frac{\sin\text{bx}}{\text{bx}}\Big)}$ $\begin{bmatrix}\text{x}\rightarrow0\Rightarrow\text{ax}\rightarrow0\\ \text{and x}\rightarrow0\Rightarrow\text{bx}\rightarrow0\end{bmatrix}$
$=\big(\frac{\text{a}}{\text{b}}\big)\times\frac{1}{1}$ $\bigg[\lim\limits_{\text{y}\rightarrow0}\frac{\sin\text{y}}{\text{y}}=1\bigg]$
$=\frac{\text{a}}{\text{b}}$

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