Answer
$$0$$
Work Step by Step
$$\eqalign{
& {\text{Calculate }}\mathop {\lim }\limits_{x \to \infty } \frac{{{{\tan }^{ - 1}}x}}{x} \cr
& {\text{Evaluate the limit}} \cr
& {\text{ }} = \mathop {\lim }\limits_{x \to \infty } \frac{{\overbrace {\left( {{{\tan }^{ - 1}}x} \right)}^{{\text{approaches to }}\frac{\pi }{2}}}}{{\underbrace {x}_{{\text{approaches to }}\infty }}} \cr
& {\text{ }} = \frac{{\frac{\pi }{2}}}{\infty } = 0 \cr
& {\text{Therefore}}{\text{,}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{{{\tan }^{ - 1}}x}}{x} = 0 \cr} $$
