Question

Evaluate
$\rm \displaystyle \lim_{x\to 0}\ \frac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a(a+x)}}$

Answer 1

To solve the given limit,rationalise the numerator

$\lim_{x\to 0}\ \frac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a(a+x)}} \times \frac{ \sqrt{a + x} + \sqrt{a} }{ \sqrt{a + x} + \sqrt{a} } \\ \\ \lim_{x\to 0}\ \frac{a + x - a}{x\sqrt{a(a+x)}} \times \frac{1}{ \sqrt{a + x} + \sqrt{a} } \\ \\ \lim_{x\to 0}\ \frac{ x }{x\sqrt{a(a+x)}} \times \frac{1}{ \sqrt{a + x} + \sqrt{a} } \\ \\ \lim_{x\to 0}\ \frac{ 1 }{\sqrt{a(a+x)}} \times \frac{1}{ \sqrt{a + x} + \sqrt{a} } \\ \\ apply \: limit \\ \\ = \frac{ 1 }{\sqrt{a(a+0)}} \times \frac{1}{ \sqrt{a + 0} + \sqrt{a} } \\ \\ = \frac{ 1 }{\sqrt{ {a}^{2} }} \times \frac{1}{ \sqrt{a } + \sqrt{a} } \\ \\ \lim_{x\to 0}\ \frac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a(a+x)}} = \frac{1}{2a \sqrt{a} }$
Hope it helps you.