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Rationalization Method to Remove Indeterminate Form

Find the limi...

Question

Find the limits of $\frac{\sqrt{x} - \sqrt{2 a} + \sqrt{x - 2 a}}{\sqrt{x^{2} - 4 a^{2}}},$ when $x = 2 a$ .

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Solution

To find the limits of $\frac{\sqrt{x} - \sqrt{2 a} + \sqrt{x - 2 a}}{\sqrt{x^{2} - 4 a^{2}}}$ when $x = 2 a$
When we apply $x = 2 a$ , we get $\frac{0}{0}$ form
So, let's simplify this

$lim x \to 2 a \frac{\sqrt{x} + \sqrt{x - 2 a} - \sqrt{2 a}}{\sqrt{x^{2} - 4 a^{2}}}$

$= lim x \to 2 a \frac{\sqrt{x} + \sqrt{x - 2 a} - \sqrt{2 a}}{\sqrt{x^{2} - 4 a^{2}}} \times \frac{(\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a})}{(\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a})}$

$= lim x \to 2 a \frac{(\sqrt{x} + \sqrt{x - 2 a})^{2} - (\sqrt{2 a})^{2})}{\sqrt{x^{2} - 4 a^{2}}} \times \frac{1}{(\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a})}$

$= lim x \to 2 a \frac{x + (x - 2 a) + 2 \sqrt{x (x - 2 a)} - 2 a}{\sqrt{x^{2} - 4 a^{2}}} \times \frac{1}{\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a}}$
$= lim x \to 2 a \frac{2 x - 4 a + 2 \sqrt{x (x - 2 a)}}{\sqrt{x^{2} - 4 a^{2}}} \times \frac{1}{\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a}}$
$= lim x \to 2 a \frac{2 (x - 2 a) + 2 \sqrt{x (x - 2 a)}}{\sqrt{(x - 2 a) (x + 2 a)}} \times \frac{1}{\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a}}$

$= lim x \to 2 a \frac{2 \sqrt{(x - 2 a)} [\sqrt{x - 2 a} + \sqrt{x}]}{\sqrt{(x - 2 a) (x + 2 a)}} \times \frac{1}{\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a}}$

$= lim x \to 2 a \frac{2 [\sqrt{x - 2 a} + \sqrt{x}]}{\sqrt{(x + 2 a)}} \times \frac{1}{\sqrt{x} + \sqrt{x - 2 a} + \sqrt{2 a}}$
$= \frac{2 [\sqrt{2 a - 2 a} + \sqrt{2 a}]}{\sqrt{(2 a + 2 a)}} \times \frac{1}{\sqrt{2 a} + \sqrt{2 a - 2 a} + \sqrt{2 a}}$

$= \frac{2 [\sqrt{2 a}]}{\sqrt{4 a}} \times \frac{1}{2 \sqrt{2 a}}$
$= \frac{1}{2 \sqrt{a}}$

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