lim _{x rightarrow a} frac{sqrt{a+2 x}-sqrt{3 | vedclass

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Question

(C) We evaluate the limit: $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$ Rationalizing the numerator and denominat

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) We evaluate the limit: $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$ Rationalizing the numerator and denominator: $= \lim _{x}$ ${\rightarrow a} \frac{(\sqrt{a+2 x}-\sqrt{3 x})(\sqrt{a+2 x}+\sqrt{3 x})(\sqrt{3 a+x}+2 \sqrt{x})}{(\sqrt{3 a+x}-2 \sqrt{x})(\sqrt{3 a+x}+2 \sqrt{x})(\sqrt{a+2 x}+\sqrt{3 x})}$ $= \lim _{x \rightarrow a} \frac{(a+2 x-3 x)(\sqrt{3 a+x}+2 \sqrt{x})}{(3 a+x-4 x)(\sqrt{a+2 x}+\sqrt{3 x})}$ $= \lim _{x \rightarrow a} \frac{(a-x)(\sqrt{3 a+x}+2 \sqrt{x})}{3(a-x)(\sqrt{a+2 x}+\sqrt{3 x})}$ Canceling $(a-x)$ and substituting $x = a$: $= \frac{\sqrt{3 a+a}+2 \sqrt{a}}{3(\sqrt{a+2 a}+\sqrt{3 a})}$ $= \frac{2 \sqrt{a}+2 \sqrt{a}}{3(\sqrt{3 a}+\sqrt{3 a})}$ $= \frac{4 \sqrt{a}}{3(2 \sqrt{3 a})} = \frac{4 \sqrt{a}}{6 \sqrt{3} \sqrt{a}} = \frac{2}{3 \sqrt{3}}$

$\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}} = $

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