Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{x \to \pi } \frac{{\tan x}}{x} \cr
& = \mathop {\lim }\limits_{x \to \pi } \frac{{\left( {\sin x/\cos x} \right)}}{x} \cr
& = \mathop {\lim }\limits_{x \to \pi } \left( {\frac{{\sin x}}{{\cos x}}} \right)\left( {\mathop {\lim }\limits_{x \to \pi } \frac{1}{x}} \right) \cr
& {\text{evaluating, we get:}} \cr
& = \left( {\frac{{\sin \pi }}{{\cos \pi }}} \right)\left( {\frac{1}{\pi }} \right) \cr
& = \left( {\frac{0}{{ - 1}}} \right)\left( {\frac{1}{\pi }} \right) \cr
& = 0 \cr} $$
