_ x 0 2 x+ 6 x 5 x- 3 x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{\sin 2 x+\sin 6 x}{\sin 5 x-\sin 3 x}=$

A
$\frac{1}{2}$
B
$\frac{1}{4}$
C
2
✓
4

✓

Answer

Correct option: D.

4

(D)
$\lim _{x \rightarrow 0} \frac{2 \sin 4 x \cos 2 x}{2 \sin x \cos 4 x}$
$=\lim _{x \rightarrow 0} 4\left(\frac{\sin 4 x}{4 x}\right)\left(\frac{x}{\sin x}\right) \frac{\cos 2 x}{\cos 4 x}=4$
Alternate method:
$\lim _{x \rightarrow 0} \frac{\frac{2 \sin 2 x}{2 x}+\frac{6 \sin 6 x}{6 x}}{\frac{5 \sin 5 x}{5 x}-\frac{3 \sin 3 x}{3 x}}=\frac{2+6}{5-3}=4$

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