limx→∞ {x+x+x−x} is equal to

A
0
B
$1 / 2$
C
$\log 2$
D
$e^{4}$

Solution:

We have,

$\begin{array}{l} lim_{x \to \infty} {\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}} \\ = lim_{x \to \infty} \frac{x + \sqrt{x + \sqrt{x}} - x}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \\ = lim_{x \to \infty} \frac{\sqrt{x + \sqrt{x}}}{\sqrt{x + \sqrt{x + \sqrt{x}} + \sqrt{x}}} \\ = lim_{x \to \infty} \frac{\sqrt{1 + x^{- 1 / 2}}}{\sqrt{1 + \sqrt{x^{- 1} + x^{- 3 / 2}}} + 1} = \frac{1}{2} \end{array}$

$lim_{x \to \infty} {\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}}$ is equal to

Solution:

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