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

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

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(A) Given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{\sin ax + bx}{ax + \sin bx}$.At $x=0$,the expression takes the indeterminate form $\frac{0}{0}$.We can divide the numerator and the denominator by $x$:$\mathop {\lim }\limits_{x 	o 0} \frac{\frac{\sin ax}{x} + b}{\frac{ax}{x} + \frac{\sin bx}{x}}$Using the standard limit $\mathop {\lim }\limits_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$,we multiply and divide by $a$ and $b$ respectively:$= \mathop {\lim }\limits_{x 	o 0} \frac{a \cdot \frac{\sin ax}{ax} + b}{a + b \cdot \frac{\sin bx}{bx}}$Applying the limit as $x 	o 0$:$= \frac{a(1) + b}{a + b(1)}$$= \frac{a+b}{a+b}$$= 1$.

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

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