If for some real number a,limx→0 sin⁡2x+asin⁡xx3exists, then the limit is equal to

If for some real number $a, lim_{x \to 0} \frac{\sin 2 x + a \sin x}{x^{3}}$
exists, then the limit is equal to

A
– 2
B
– 1
C
1
D
2

Solution:

$lim_{x \to 0} \frac{\sin 2 x + a \sin x}{x^{3}} = lim_{x \to 0} \frac{2 \cos 2 x + a \cos x}{3 x^{2}} (\frac{0}{0} form)$

The last limits will exist if $a = - 2$ In this case the
last limit is equal to

$lim_{x \to 0} \frac{- 4 \sin 2 x + 2 \sin x}{6 x} = lim_{x \to 0} [- \frac{4}{3} \frac{\sin 2 x}{2 x} + \frac{1}{3} \frac{\sin x}{x}]$

$= - \frac{4}{3} + \frac{1}{3} = - 1$

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