Question

The value of $\underset{x\to \infty }{lim}\sqrt{x}(\sqrt{x+c}-\sqrt{x})$ is

• A. $\frac{c}{2}$
• B. $\frac{c}{3}$
• C. $\frac{c}{4}$
• D. none of these

Answer 1

Answer: (A)

$\frac{c}{2}$

$\underset{x\to \infty }{lim}\sqrt{x}(\sqrt{x+c}-\sqrt{x})$
$=\underset{x\to \infty }{lim}\sqrt{x}(\sqrt{x+c}-\sqrt{x}).\frac{\sqrt{x+c}+\sqrt{x}}{\sqrt{x+c}+\sqrt{x}}$
$=\underset{x\to \infty }{lim}\frac{c\sqrt{x}}{\sqrt{x+c}+\sqrt{x}}$
$=\underset{x\to \infty }{lim}\frac{c}{\sqrt{1+c/x}+1}=\frac{c}{1+1}=\frac{c}{2}$