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

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

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$\frac{a}{b}$

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(A) We are given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax}{bx}$At $x=0$,the function takes the indeterminate form $\frac{0}{0}$.To evaluate this,we use the standard limit formula $\mathop {\lim }\limits_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$.Multiply and divide by $a$:$\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax}{bx} = \mathop {\lim }\limits_{x 	o 0} \left( \frac{\sin ax}{ax} 	imes \frac{ax}{bx} ight)$$= \mathop {\lim }\limits_{x 	o 0} \left( \frac{\sin ax}{ax} ight) 	imes \frac{a}{b}$Since $x 	o 0$,it follows that $ax 	o 0$. Therefore:$= \frac{a}{b} 	imes \mathop {\lim }\limits_{ax 	o 0} \left( \frac{\sin ax}{ax} ight)$$= \frac{a}{b} 	imes 1 = \frac{a}{b}$

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} \frac{\sin ax}{bx}$

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