Answer
$${\text{the improper integral diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^5 {\frac{{10}}{x}} dx \cr
& \frac{{10}}{x}{\text{ has an infinite discontinuity at }}x = 0,{\text{we can write}} \cr
& \int_0^5 {\frac{{10}}{x}} dx = \mathop {\lim }\limits_{b \to {0^ + }} \int_b^5 {\frac{{10}}{x}} dx \cr
& {\text{Integrate}} \cr
& = \mathop {\lim }\limits_{b \to {0^ + }} \left[ {10\ln \left| x \right|} \right]_b^5 \cr
& = 10\mathop {\lim }\limits_{b \to {0^ + }} \left[ {\ln 5 - \ln \left| b \right|} \right] \cr
& {\text{Evaluate the limit}} \cr
& = 10\left( {\ln 5 - \ln \left| {{0^ + }} \right|} \right) \cr
& = 10\left( {\ln 5 - \infty } \right) \cr
& = - \infty \cr
& {\text{So}},{\text{ the improper integral diverges}}. \cr
& {\text{The following graph confirms the result}} \cr} $$
