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

$$\stackrel{@\mathrm{bemath}@}{}$$$$\underset{x\to 0}{\lim }\frac{\sqrt{1+2\sin \mathrm{x}}-\sqrt{1-4\sin 4\mathrm{x}}}{4\mathrm{x}}$$

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

$$\underset{@\mathrm{JS}@}{}$$$$\underset{x\to 0}{\lim }\frac{(1+\frac{2\sin \mathrm{x}}{2})-(1-\frac{4\sin 4\mathrm{x}}{2})}{4\mathrm{x}}=$$$$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}+2\sin 4\mathrm{x}}{4\mathrm{x}}=\frac{9}{4}$$

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

$$\underset{x\to 0}{\lim }\frac{1+sinx-1+2sin4x}{4x}$$$$\underset{x\to 0}{\lim }\frac{x+8x}{4x}=\frac{9}{4}sinx\to x,sin4x\to 4x$$

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

$$\underset{x\to 0}{\lim }\frac{1+2sinx-1+4sin4x}{4x}.\frac{1}{\sqrt{1+2sinx}+\sqrt{1-4sin4x}}$$$$\underset{x\to 0}{\lim }\frac{1}{4}(\frac{2sinx}{x}+\frac{16sin4x}{4x}).\frac{1}{2}$$$$\frac{1}{4}(18.\frac{1}{2})=\frac{9}{4}$$

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