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

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Question

(D) To evaluate the limit $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 a}}{\sqrt{x}-\sqrt{a}}$,we rationalize the numerator and the denominator

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) To evaluate the limit $\lim _{x ightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 a}}{\sqrt{x}-\sqrt{a}}$,we rationalize the numerator and the denominator: $\lim _{x}$ ${ightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 a}}{\sqrt{x}-\sqrt{a}} 	imes \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}} 	imes \frac{\sqrt{a+2 x}+\sqrt{3 a}}{\sqrt{a+2 x}+\sqrt{3 a}}$ $= \lim _{x ightarrow a} \frac{(a+2x) - 3a}{x-a} 	imes \frac{\sqrt{x}+\sqrt{a}}{\sqrt{a+2x}+\sqrt{3a}}$ $= \lim _{x ightarrow a} \frac{2x - 2a}{x-a} 	imes \frac{\sqrt{x}+\sqrt{a}}{\sqrt{a+2x}+\sqrt{3a}}$ $= \lim _{x ightarrow a} \frac{2(x-a)}{x-a} 	imes \frac{\sqrt{x}+\sqrt{a}}{\sqrt{a+2x}+\sqrt{3a}}$ $= 2 	imes \frac{\sqrt{a}+\sqrt{a}}{\sqrt{a+2a}+\sqrt{3a}} = 2 	imes \frac{2\sqrt{a}}{\sqrt{3a}+\sqrt{3a}} = 2 	imes \frac{2\sqrt{a}}{2\sqrt{3a}} = \frac{2}{\sqrt{3}}$

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

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