Find the function’s end behavior as (x to infty) and (x to -infty).
If (f(x)) approaches a constant (L), then the horizontal asymptote is (y=L).
For rational functions (frac{p(x)}{q(x)}):
If (deg(p) < deg(q)), the horizontal asymptote is (y=0).
If (deg(p) = deg(q)), the horizontal asymptote is (y=) (leading coefficient of (p)) (/) (leading coefficient of (q)).
If (deg(p) > deg(q)), there is no horizontal asymptote.
Use the limit definition:
Horizontal asymptote as (x to infty): (y=lim_{xtoinfty} f(x)) (if the limit exists and is finite).
Horizontal asymptote as (x to -infty): (y=lim_{xto-infty} f(x)) (if the limit exists and is finite).
If the two one-sided limits are different finite constants, there are two different horizontal asymptotes (one for each direction).
If either limit is infinite or does not exist, there is no horizontal asymptote.