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Question

Question: \(\lim_{x \rightarrow \infty}\left\lbrack \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right\rbrack\)i...

limx[x+x+xx]\lim_{x \rightarrow \infty}\left\lbrack \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right\rbrackis equal to

A

0

B

12\frac{1}{2}

C

log2\log 2

D

e4e^{4}

Answer

12\frac{1}{2}

Explanation

Solution

limx[x+x+xx]\lim_{x \rightarrow \infty}\left\lbrack \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right\rbrack=limxx+x+xxx+x+x+x\lim_{x \rightarrow \infty}\frac{x + \sqrt{x + \sqrt{x}} - x}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}}

=limxx+xx+x+x+x=limx1+x1/21+x1+x3/2+1=12\lim_{x \rightarrow \infty}\frac{\sqrt{x + \sqrt{x}}}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} = \lim_{x \rightarrow \infty}\frac{\sqrt{1 + x^{- 1/2}}}{\sqrt{1 + \sqrt{x^{- 1} + x^{- 3/2}}} + 1} = \frac{1}{2}.