\(
\eqalign{
& \mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {x + 6} - \sqrt {3x} }}
{{\sqrt {x - 3} }} = \cr
& \mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {x + 6} - \sqrt {3x} }}
{{\sqrt {x - 3} }} \cdot \frac{{\sqrt {x + 6} + \sqrt {3x} }}
{{\sqrt {x + 6} + \sqrt {3x} }} = \cr
& \mathop {\lim }\limits_{x \to 3} \frac{{x + 6 - 3x}}
{{\sqrt {x - 3} \cdot \left( {\sqrt {x + 6} + \sqrt {3x} } \right)}} = \cr
& \mathop {\lim }\limits_{x \to 3} \frac{{ - 2x + 6}}
{{\sqrt {x - 3} \cdot \left( {\sqrt {x + 6} + \sqrt {3x} } \right)}} = \cr
& \mathop {\lim }\limits_{x \to 3} \frac{{ - 2\left( {x - 3} \right)}}
{{\sqrt {x - 3} \cdot \left( {\sqrt {x + 6} + \sqrt {3x} } \right)}} = \cr
& \mathop {\lim }\limits_{x \to 3} \frac{{ - 2\sqrt {x - 3} }}
{{\sqrt {x + 6} + \sqrt {3x} }} = \frac{{ - 2\sqrt {3 - 3} }}
{{\sqrt {3 + 6} + \sqrt {3 \cdot 3} }} = 0 \cr}
\)
