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Question: \(\lim_{x \rightarrow a}\frac{\sqrt{a + 2x} - \sqrt{3x}}{\sqrt{3a + x} - 2\sqrt{x}}\) equals...

lim⁡x→aa+2x−3x3a+x−2x\lim_{x \rightarrow a}\frac{\sqrt{a + 2x} - \sqrt{3x}}{\sqrt{3a + x} - 2\sqrt{x}} equals

A

2a33\frac{2a}{3\sqrt{3}}

B

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

C

0

D

None of these

Answer

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

Explanation

Solution

lim⁡x→aa+2x−3x3a+x−2x=lim⁡x→a(a+2x−3x3a+x−2x)×(a+2x+3xa+2x+3x)×(3a+x+2x3a+x+2x)=lim⁡x→a{3a+x+2x3(a+2x+3x)}=233\lim_{x \rightarrow a}\frac{\sqrt{a + 2x} - \sqrt{3x}}{\sqrt{3a + x} - 2\sqrt{x}} = \lim_{x \rightarrow a}\left( \frac{\sqrt{a + 2x} - \sqrt{3x}}{\sqrt{3a + x} - 2\sqrt{x}} \right) \times \left( \frac{\sqrt{a + 2x} + \sqrt{3x}}{\sqrt{a + 2x} + \sqrt{3x}} \right) \times \left( \frac{\sqrt{3a + x} + 2\sqrt{x}}{\sqrt{3a + x} + 2\sqrt{x}} \right) = \lim_{x \rightarrow a}\left\{ \frac{\sqrt{3a + x} + 2\sqrt{x}}{3(\sqrt{a + 2x} + \sqrt{3x})} \right\} = \frac{2}{3\sqrt{3}}.