Horizontal Asymptotes,
Slant Asymptotes

A function has either a horizontal or a vertical asymptote, but not both. Unlike vertical asymptotes, horizontal asymptotes are just guides and you're allowed to cross them.

If the degree of the denominator is larger, there's a horizontal asymptote at y=0.



To find vertical asymptotes, set the denominator equal to zero.



This can't be factored. The quadratic formula shows there are only imaginary solutions. So there are no vertical asymptotes.

---

If the degrees are equal, there's also a horizontal asymptote.



Find it by dividing the leading terms.



So there's a horizontal asymptote at y=1.

Vertical asymptotes:




So there are vertical asymptotes at x=2 and x=-3/2.



(Notice the open circle at x=2, indicating that the function is undefined there.)

---

If the degree of the numerator is larger, there's a slant asymptote.



Distribute and put it in order.



Use long division.



Throw away the remainder.
There's a slant asymptote at:


Vertical asymptotes:



So there's a vertical asymptote at x=-4.

---