Evaluate x - alim-a - 2x - -3x-3a - x -2 -x- a - 0

The correct option is A 23 √(3) lim_x→ a√(a+2x) √(3x)√(3a+x) 2√(x) lim_x→ a√(a+2x) √(3x)√(a+2x)+√(3x)√(a+2x)+√(3x)1√(3a+x) 2√(x)√(3a+x)+2√(x)√(3 ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

Evaluate x ...

Question

Evaluate $l i m x \to a \frac{\sqrt{a + 2 x} - \sqrt{3 x}}{\sqrt{3 a + x} - 2 \sqrt{x}}, (a \neq 0)$

\frac{2}{3 \sqrt{3}}

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\frac{4}{3 \sqrt{3}}

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\frac{5}{3 \sqrt{3}}

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\frac{7}{3 \sqrt{3}}

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Solution

The correct option is A $\frac{2}{3 \sqrt{3}}$
${lim}_{x \to a} \frac{\sqrt{a + 2 x} - \sqrt{3 x}}{\sqrt{3 a + x} - 2 \sqrt{x}}$

${lim}_{x \to a} \sqrt{a + 2 x} - \sqrt{3 x} \frac{\sqrt{a + 2 x} + \sqrt{3 x}}{\sqrt{a + 2 x} + \sqrt{3 x}} \frac{1}{\sqrt{3 a + x} - 2 \sqrt{x}} \frac{\sqrt{3 a + x} + 2 \sqrt{x}}{\sqrt{3 a + x} + 2 \sqrt{x}}$

${lim}_{x \to a} \frac{[(a + 2 x) - 3 x]}{[(3 a + x) - 4 x]} \cdot \frac{\sqrt{3 a + x} + 2 \sqrt{x}}{\sqrt{a + 2 x} + \sqrt{3 x}}$

${lim}_{x \to a} \frac{(a - x)}{(3 a - 3 x)} \cdot \frac{\sqrt{3 a + x} + 2 \sqrt{x}}{\sqrt{a + 2 x} + \sqrt{3 x}}$

${lim}_{x \to a} \frac{1}{3} \cdot \frac{\sqrt{3 a + x} + 2 \sqrt{x}}{\sqrt{a + 2 x} + \sqrt{3 x}}$

$= \frac{1}{3} \frac{(\sqrt{4 a} + 2 \sqrt{a})}{(\sqrt{3 a} + \sqrt{3 a})}$

$= \frac{1}{3} \cdot \frac{2}{\sqrt{3}}$

$= \frac{2}{3 \sqrt{3}}$

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Evaluate limx→a√a+2x−√3x√3a+x−2√x,(a≠0)

Evaluate $l i m x \to a \frac{\sqrt{a + 2 x} - \sqrt{3 x}}{\sqrt{3 a + x} - 2 \sqrt{x}}, (a \neq 0)$