_ x a a + 2x - 3x 3a + x - 2 x = (a 0) - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to a} \frac{{\sqrt {a + 2x} - \sqrt {3x} }}{{\sqrt {3a + x} - 2\sqrt x }} = $$(a \ne 0)$

A
$\frac{1}{{\sqrt 3 }}$
✓
$\frac{2}{{3\sqrt 3 }}$
C
$\frac{2}{{\sqrt 3 }}$
D
$\frac{2}{3}$

✓

Answer

Correct option: B.

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

b
(b) $\mathop {\lim }\limits_{x \to a} \,\frac{{\sqrt {a + 2x} - \sqrt {3x} }}{{\sqrt {3a + x} - 2\sqrt x }}$

$ = \mathop {\lim }\limits_{x \to a} \,\frac{{\sqrt {a + 2x} - \sqrt {3x} }}{{\sqrt {3a + x} - 2\sqrt x }} \times \frac{{\sqrt {a + 2x} + \sqrt {3x} }}{{\sqrt {a + 2x} + \sqrt {3x} }} \times \frac{{\sqrt {3a + x} + 2\sqrt x }}{{\sqrt {3a + x} + 2\sqrt x }}$

$ = \mathop {\lim }\limits_{x \to a} \frac{{\sqrt {3a + x} + 2\sqrt x }}{{3\,(\sqrt {a + 2x} + \sqrt {3x)} }} = \frac{2}{{3\sqrt 3 }}$.

Aliter : Apply $L-$ Hospital’s rule.

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