lim x → 0 2 sin x − sin 2 x x 3 is equal to Options: A . &nbsp1 B . &nbsp−1 C . &nbsp0 D . &nbspdoes not exist

Answer: Option A : A lim x → 0 2 sin x − sin 2 x x 3 = lim x → 0 2 sin x ( 1 − cos x ) ( 1 + cos x ) x 3 ( 1 + cos x ) = lim x → 0 2 sin 3 x x 3 × 1 1 + cos x = 2 × ( 1 ) 3 × 1 1 + 1 = 1

limx→02sinx−sin2xx3 is equal to

Question

lim x \to 0 \frac{2 sin x - sin 2 x}{x^{3}}

is equal to

Options:

A . 1

B . −1

C . 0

D . does not exist

Answer: Option A
:
A

lim x \to 0 \frac{2 sin x - sin 2 x}{x^{3}}

= lim x \to 0 \frac{2 sin x (1 - cos x) (1 + cos x)}{x^{3} (1 + cos x)}

= lim x \to 0 2 \frac{{sin}^{3} x}{x^{3}} \times \frac{1}{1 + cos x}

= 2 \times (1)^{3} \times \frac{1}{1 + 1} = 1

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