$\mathop {\lim }\limits_{x \to 2} \frac{{x - \sqrt {x + 2} }}{{{x^2} + x - 6}} = \mathop {\lim }\limits_{x \to 2} \frac{{x - \sqrt {x + 2} }}{{{x^2} + x - 6}} \times \frac{{x + \sqrt {x + 2} }}{{x + \sqrt {x + 2} }} = \mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - (x + 2)}}{{(x + 3)(x - 2)(x + \sqrt {x + 2} )}} = $
$\mathop {\lim }\limits_{x \to 2} \frac{{(x - 2)(x + 1)}}{{(x + 3)(x - 2)(x + \sqrt {x + 2} )}} = \mathop {\lim }\limits_{x \to 2} \frac{{(x + 1)}}{{(x + 3)(x + \sqrt {x + 2} )}} = \frac{3}{{20}}$
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