Can you cross a slant asymptote?
Correspondingly, can you cross an asymptote?
Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior, usually far off to the sides of the graph.
Correspondingly, how do you find the crossing point of a slant asymptote?
Your oblique asymptote equation is correct, but your work is wrong. You should get x=1 as your x coordinate for the point of intersection. To find the y coordinate, simply plug in x=1 to either equation and you'll see that the point of intersection is (1,0).
If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote. If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote.