The value of lim _{x rightarrow a} frac{sqrt{ | vedclass

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(C) We evaluate the limit $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$ by rationalizing both the numerator and th

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

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(C) We evaluate the limit $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$ by rationalizing both the numerator and the denominator. Multiplying by the conjugates: $\lim _{x}$ ${\rightarrow a} \frac{(\sqrt{a+2 x}-\sqrt{3 x})(\sqrt{a+2 x}+\sqrt{3 x})(\sqrt{3 a+x}+2 \sqrt{x})}{(\sqrt{3 a+x}-2 \sqrt{x})(\sqrt{3 a+x}+2 \sqrt{x})(\sqrt{a+2 x}+\sqrt{3 x})}$ Simplifying the terms using $(A-B)(A+B) = A^2 - B^2$: $\lim _{x \rightarrow a} \frac{(a+2 x-3 x)(\sqrt{3 a+x}+2 \sqrt{x})}{(3 a+x-4 x)(\sqrt{a+2 x}+\sqrt{3 x})}$ $\lim _{x \rightarrow a} \frac{(a-x)(\sqrt{3 a+x}+2 \sqrt{x})}{3(a-x)(\sqrt{a+2 x}+\sqrt{3 x})}$ Canceling the common factor $(a-x)$: $\lim _{x \rightarrow a} \frac{\sqrt{3 a+x}+2 \sqrt{x}}{3(\sqrt{a+2 x}+\sqrt{3 x})}$ Substituting $x = a$: $\frac{\sqrt{4a} + 2\sqrt{a}}{3(\sqrt{3a} + \sqrt{3a})} = \frac{2\sqrt{a} + 2\sqrt{a}}{3(2\sqrt{3a})} = \frac{4\sqrt{a}}{6\sqrt{3a}} = \frac{2}{3\sqrt{3}}$

The value of $\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$ is

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