Question:
Evaluate the following limits:
$\lim _{x \rightarrow 0} \frac{\sin 2 x+\sin 3 x}{2 x+\sin 3 x}$
Solution:
$=\lim _{x \rightarrow 0} \frac{\sin 2 x+\sin 3 x}{2 x+\sin 3 x} \times \frac{3 x}{3 x}$
$=\lim _{x \rightarrow 0} \frac{\frac{\sin 2 x+\sin 3 x}{3 x}}{\frac{2 x+\sin 3 x}{3 x}}$
$=\frac{\frac{2}{3}+1}{\frac{2}{3}+1}$
$=1$
$\therefore \lim _{x \rightarrow 0} \frac{\sin 2 x+\sin 3 x}{2 x+\sin 3 x}=1$
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