Evaluate the given limit: mathop {lim }limits | vedclass

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(A) We are given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax}{\sin bx}$.At $x=0$,the expression takes the indeterminate form $\frac{0}{

Vedclass · Accepted Answer

$\frac{a}{b}$

Vedclass · Answer

(A) We are given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax}{\sin bx}$.At $x=0$,the expression takes the indeterminate form $\frac{0}{0}$.We use the standard limit $\mathop {\lim }\limits_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$.$\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax}{\sin bx} = \mathop {\lim }\limits_{x 	o 0} \left( \frac{\sin ax}{ax} \cdot ax \cdot \frac{1}{\frac{\sin bx}{bx} \cdot bx} ight)$$= \mathop {\lim }\limits_{x 	o 0} \left( \frac{\sin ax}{ax} ight) \cdot \frac{1}{\mathop {\lim }\limits_{x 	o 0} \left( \frac{\sin bx}{bx} ight)} \cdot \frac{ax}{bx}$$= 1 \cdot \frac{1}{1} \cdot \frac{a}{b} = \frac{a}{b}$.

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} \frac{\sin ax}{\sin bx}$,where $a, b \neq 0$.

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