De worteltruuk

$\eqalign{ & \mathop {\lim }\limits_{x \to 3} \frac{{5 - \sqrt {x^2 + 16} }} {{3 - x}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {x^2 + 16} - 5}} {{x - 3}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {x^2 + 16} - 5}} {{x - 3}} \cdot \frac{{\sqrt {x^2 + 16} + 5}} {{\sqrt {x^2 + 16} + 5}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{x^2 + 16 - 25}} {{\left( {x - 3} \right)\left( {\sqrt {x^2 + 16} + 5} \right)}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{x^2 - 9}} {{\left( {x - 3} \right)\left( {\sqrt {x^2 + 16} + 5} \right)}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{\left( {x - 3} \right)\left( {x + 3} \right)}} {{\left( {x - 3} \right)\left( {\sqrt {x^2 + 16} + 5} \right)}} = \cr & \mathop {\lim }\limits_{x \to 3} \frac{{x + 3}} {{\sqrt {x^2 + 16} + 5}} = \frac{6} {{\sqrt {3^2 + 16} + 5}} = \frac{3} {5} \cr}$

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