bemath-lim-x-0-1-2sin-x-1-4sin-4x-4x-

Question Number 106633 by bemath last updated on 06/Aug/20

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\lim_{x \to 0} \frac{\sqrt{1 + 2 \sin x} - \sqrt{1 - 4 \sin 4 x}}{4 x}

Answered by john santu last updated on 06/Aug/20

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\lim_{x \to 0} \frac{(1 + \frac{2 \sin x}{2}) - (1 - \frac{4 \sin 4 x}{2})}{4 x} =

\lim_{x \to 0} \frac{\sin x + 2 \sin 4 x}{4 x} = \frac{9}{4}

Answered by Dwaipayan Shikari last updated on 06/Aug/20

\lim_{x \to 0} \frac{1 + s i n x - 1 + 2 s i n 4 x}{4 x}

\lim_{x \to 0} \frac{x + 8 x}{4 x} = \frac{9}{4} s i n x \to x, s i n 4 x \to 4 x

Answered by Dwaipayan Shikari last updated on 06/Aug/20

\lim_{x \to 0} \frac{1 + 2 s i n x - 1 + 4 s i n 4 x}{4 x} . \frac{1}{\sqrt{1 + 2 s i n x} + \sqrt{1 - 4 s i n 4 x}}

\lim_{x \to 0} \frac{1}{4} (\frac{2 s i n x}{x} + \frac{16 s i n 4 x}{4 x}) . \frac{1}{2}

\frac{1}{4} (18 . \frac{1}{2}) = \frac{9}{4}