In what ways can horizontal, vertical, and oblique asymptotes be identified?

Vertical asymptotes can be defined by looking for gaps in the domain. In the equation f(x) = 1/x, you cannot put 0 into the equation. This is because we can't divide by 0. Giving us a vertical asymptote.  

Horizontal asymptotes can be found by using the holes in the domain to find the range values that are not possible. In the same equation as above, there is no way to get a value of y = 0. Thus, you have a horizontal asymptote. 

Oblique asymptotes are a bit harder to find. For this you would have to have a higher degree polynomial in the numerator than the denomination. In this case you would use synthetic division to find the slant. Example: 5x^3/x^2 + 5.