lim_x→ 3x 3√(2 x) √(4 x) =lim_x→ 3x 3√(2 x) √(4 x)×√(2 x)+√(4 x)√(2 x)+√(4 x) =lim_x→ 3(x 3)(√(2 x)+√(4 x))2 x 4+x =lim_x→ 3(x 3)(√(2 x)+√(4 ...

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Rationalization Method to Remove Indeterminate Form

lim x→ 3 x...

Question

$\begin{matrix} l i m x \to 3 \end{matrix} \frac{x - 3}{\sqrt{2 - x} - \sqrt{4 - x}}$

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Solution

${lim}_{x \to 3} \frac{x - 3}{\sqrt{2 - x} - \sqrt{4 - x}}$
$= {lim}_{x \to 3} \frac{x - 3}{\sqrt{2 - x} - \sqrt{4 - x}} \times \frac{\sqrt{2 - x} + \sqrt{4 - x}}{\sqrt{2 - x} + \sqrt{4 - x}}$
$= {lim}_{x \to 3} \frac{(x - 3) (\sqrt{2 - x} + \sqrt{4 - x})}{2 - x - 4 + x}$
$= {lim}_{x \to 3} \frac{(x - 3) (\sqrt{2 - x} + \sqrt{4 - x})}{- 2}$
$= \frac{(3 - 3) (\sqrt{2 - 3} + \sqrt{4 - 3})}{- 2}$
$= 0$

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