Answer
$${\text{diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^4 {\frac{{dx}}{{{{\left( {x - 4} \right)}^2}}}} \cr
& {\text{The function is undefined for }}x = 4,{\text{ so the integral can be represented as}} \cr
& \int_0^4 {\frac{{dx}}{{{{\left( {x - 4} \right)}^2}}}} = \mathop {\lim }\limits_{k \to {4^ - }} \int_0^k {\frac{{dx}}{{{{\left( {x - 4} \right)}^2}}}} \cr
& {\text{Integrate}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to {4^ - }} \left[ { - \frac{1}{{x - 4}}} \right]_0^k \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to {4^ - }} \left[ { - \frac{1}{{k - 4}} + \frac{1}{{0 - 4}}} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to {4^ - }} \left[ { - \frac{1}{{k - 4}} - \frac{1}{4}} \right] \cr
& \,{\text{Calculate the limit when }}k \to {4^ - } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{{{0^ - }}} - \frac{1}{4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = + \infty \cr
& {\text{Then}}{\text{,}} \cr
& \int_0^4 {\frac{{dx}}{{{{\left( {x - 4} \right)}^2}}}} = + \infty \cr
& {\text{The integral diverges}} \cr} $$
