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

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$\frac{c}{4}$

$\frac{c}{3}$

$\frac{c}{2}$

$\frac{c}{5}$

answer is A.

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Detailed Solution

$\lim_{x \to \infty} \sqrt{x} (\sqrt{x + c} - \sqrt{x})$

$= \lim_{x \to \infty} \sqrt{x} (\sqrt{x + c} - \sqrt{x}) \times \frac{(\sqrt{x + c} + \sqrt{x})}{(\sqrt{x + c} + \sqrt{x})}$

$= \lim_{x \to \infty} \frac{\sqrt{x} (x + c - x)}{(\sqrt{x + c} + \sqrt{x})}$

$= \lim_{x \to \infty} \frac{c \sqrt{x}}{\sqrt{x + c} + \sqrt{x}}$

$= \lim_{x \to \infty} \frac{c}{\sqrt{1 + \frac{c}{x}} + 1}$ $= \frac{c}{2}$

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