Graphing rational functions can be quite the challenge as there are three possible ways of doing it based on the degree of the numerator compared to the degree of the denominator. In the example you see below the degree of the numerator is one more than the degree of the denominator.
When this occurs, one can perform a long division to determine the oblique or slant asymptote. The quotient from the long division is the slant asymptote which in this particular case is x+5. Also x=3 is the vertical asymptote since 3 is the value that makes the denominator equal zero.
Also one can see from the graph that it never crosses or intersects the slant asymptote. You can prove this by setting x+5 equal to the rational function and seeing there is no solution.