Evaluate the following limits


Evaluate the following limits

$\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x}$, where $a, b, a+b \neq 0$



$=\lim _{x \rightarrow 0} \frac{\sin (a x)+b x}{a x+\sin (b x)}$

$=\lim _{x \rightarrow 0} \frac{\sin (a x)+b x}{a x+\sin (b x)} \times \frac{b x}{a x} \times \frac{a}{b}$

$=\lim _{x \rightarrow 0} \frac{\frac{\sin a x+b x}{a x}}{\frac{a x+\sin b x}{b x}} \times \frac{a}{b}$

$=\frac{a}{b} \times \frac{\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x}}{\lim _{x \rightarrow 0} \frac{a x+\sin b x}{b x}}$

$=\frac{a}{b} \times \frac{1+\frac{b}{a}}{1+\frac{a}{b}}$

$=\frac{\mathrm{a}}{\mathrm{b}} \times \frac{\mathrm{b}}{\mathrm{a}}$


$\therefore \lim _{x \rightarrow 0} \frac{\sin (a x)+b x}{a x+\sin (b x)}=1$


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