Answer
$${\text{diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^\infty {\cos xdx} \cr
& {\text{definition of improper integral}} \cr
& \int_0^\infty {\cos xdx} = \mathop {\lim }\limits_{b \to \infty } \int_0^b {\cos xdx} \cr
& {\text{evaluate the integral}} \cr
& = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin x} \right)} \right|_0^b \cr
& = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin b - \sin 0} \right)} \right| \cr
& = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin b} \right)} \right| \cr
& {\text{evaluate the limit}} \cr
& = \sin \left( \infty \right) \cr
& {\text{the limit does not exist}}{\text{, the integral diverges}} \cr
& {\text{diverges}} \cr} $$
