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