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Question Number 78324 by jagoll last updated on 16/Jan/20

$$\underset{x\to 0}{\lim }\frac{1+\sin 4x-\cos 2x}{\sqrt{x+1}-\sqrt{1-x}}=$$

Commented by john santu last updated on 16/Jan/20

$$\underset{x\to 0}{\lim }(\sqrt{x+1}+\sqrt{1-x})\times \underset{x\to 0}{\lim }\frac{2\sin {}^{2}x+2\sin 2x\cos 2x}{2x}$$$$=2\times \underset{x\to 0}{\lim }\frac{2\sin x\{\sin x+2\cos x\cos 2x\}}{2x}$$$$=2\times \left\{2\right\}=4$$

Commented by mathmax by abdo last updated on 16/Jan/20

$$letf\left(x\right)=\frac{1+sin\left(4x\right)-cos\left(2x\right)}{\sqrt{1+x}-\sqrt{1-x}}wehave$$$$\sqrt{1+x}\sim 1+\frac{x}{2}and\sqrt{1-x}\sim 1-\frac{x}{2}alsoson\left(4x\right)\sim 4x$$$$cos\left(2x\right)\sim 1-2x^2(x\to 0)\Rightarrow f\left(x\right)\sim \frac{1+4x-1+2x^2}{1+\frac{x}{2}-1+\frac{x}{2}}$$$$\Rightarrow f\left(x\right)\sim \frac{2x^2+4x}{x}\Rightarrow f\left(x\right)\sim 2x+4\Rightarrow {lim}_{x\to 0}f\left(x\right)=4$$

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