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lt. x->a root a+2x - root3x / root3a+x - 2rootx

$\lim_{x \to a} \frac{\sqrt{a} + 2 x - \sqrt{3 x}}{\sqrt{3 a} + x - 2 \sqrt{x}} = \frac{\sqrt{a} + 2 a - \sqrt{3 a}}{\sqrt{3 a} + a - 2 \sqrt{a}} = \frac{2 a + (\sqrt{a} - \sqrt{3 a})}{a + (\sqrt{3 a} - 2 \sqrt{a})} \times \frac{a - (\sqrt{3 a} - 2 \sqrt{a})}{a - (\sqrt{3 a} - 2 \sqrt{a})} = \frac{[2 a + (\sqrt{a} - \sqrt{3 a})] [a - (\sqrt{3 a} - 2 \sqrt{a})]}{{(a)}^{2} - {(\sqrt{3 a} - 2 \sqrt{a})}^{2}} = \frac{2 a^{2} - 2 a (\sqrt{3 a} - 2 \sqrt{a}) + a (\sqrt{a} - \sqrt{3 a}) - (\sqrt{a} - \sqrt{3 a}) (\sqrt{3 a} - 2 \sqrt{a})}{a^{2} - (3 a + 4 a - 4 \sqrt{3} a)} = \frac{2 a^{2} - 2 a \sqrt{3 a} + 4 a \sqrt{a} + a \sqrt{a} - a \sqrt{3 a} - a \sqrt{3} + 2 a + 3 a - 2 a \sqrt{3}}{a^{2} - (7 a - 4 \sqrt{3} a)} = \frac{2 a^{2} - 3 a \sqrt{3 a} + 5 a \sqrt{a} + 5 a - 3 a \sqrt{3}}{a^{2} - 7 a + 4 \sqrt{3} a} = \frac{2 a - 3 \sqrt{3 a} + 5 \sqrt{a} + 5 - 3 \sqrt{3}}{a - 7 + 4 \sqrt{3}} H e n c e, \lim_{x \to a} \frac{\sqrt{a} + 2 x - \sqrt{3 x}}{\sqrt{3 a} + x - 2 \sqrt{x}} = \frac{2 a - 3 \sqrt{3 a} + 5 \sqrt{a} + 5 - 3 \sqrt{3}}{a - 7 + 4 \sqrt{3}}$

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