Answer
$$\frac{3}{4}\pi $$
Work Step by Step
$$\eqalign{
& {\text{Let the limit }}\mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right)}}{{x - 5}} \cr
& {\text{Rewrite the expression}} \cr
& \mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right)}}{{x - 5}} = \mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right) - \overbrace {\tan \left( {\pi \sqrt {3\left( 5 \right) - 11} } \right)}^{{\text{equal to 0}}}}}{{x - 5}} \cr
& = \mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right) - \tan \left( {2\pi } \right)}}{{x - 5}} \cr
& {\text{using the definition of the derivative }}f'\left( a \right) = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}} \cr
& \underbrace {\mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right) - \tan \left( {2\pi } \right)}}{{x - 5}}}_{\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}} \cr
& \Rightarrow f\left( x \right) = \tan \left( {\pi \sqrt {3x - 11} } \right){\text{ }}and{\text{ }}a = 5 \cr
& {\text{With the definition of derivative:}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan \left( {\pi \sqrt {3x - 11} } \right)} \right] \cr
& f'\left( x \right) = {\sec ^2}\left( {\pi \sqrt {3x - 11} } \right)\frac{d}{{dx}}\left[ {\pi \sqrt {3x - 11} } \right] \cr
& f'\left( x \right) = \pi {\sec ^2}\left( {\pi \sqrt {3x - 11} } \right)\left( {\frac{3}{{2\sqrt {3x - 11} }}} \right) \cr
& {\text{At }}x = 5 \cr
& f'\left( 5 \right) = \pi {\sec ^2}\left( {\pi \sqrt {3\left( 5 \right) - 11} } \right)\left( {\frac{3}{{2\sqrt {3\left( 5 \right) - 11} }}} \right) \cr
& f'\left( 5 \right) = \pi {\sec ^2}\left( {2\pi } \right)\left( {\frac{3}{4}} \right) \cr
& f'\left( 5 \right) = \pi \left( 1 \right)\left( {\frac{3}{4}} \right) \cr
& f'\left( 5 \right) = \frac{3}{4}\pi \cr
& {\text{Therefore}}{\text{,}} \cr
& \mathop {\lim }\limits_{x \to 5} \frac{{\tan \left( {\pi \sqrt {3x - 11} } \right)}}{{x - 5}} = \frac{3}{4}\pi \cr} $$
