Limits at infinity are used to describe the behavior of functions as the independent variable increases or decreases without bound. If a function approaches a numerical value L in either of these situations, write
and f( x) is said to have a horizontal asymptote at y = L. A function may have different horizontal asymptotes in each direction, have a horizontal asymptote in one direction only, or have no horizontal asymptotes.
Evaluate 1: Evaluate 
Factor the largest power of x in the numerator from each term and the largest power of x in the denominator from each term.
You find that
The function has a horizontal asymptote at y = 2.
Example 2: Evaluate 
Factor x 3 from each term in the numerator and x 4 from each term in the denominator, which yields
The function has a horizontal asymptote at y = 0.
Example 3: Evaluate 
.
Factor x 2 from each term in the numerator and x from each term in the denominator, which yields
Because this limit does not approach a real number value, the function has no horizontal asymptote as x increases without bound.
Example 4: Evaluate 
.
Factor x 3 from each term of the expression, which yields
As in the previous example, this function has no horizontal asymptote as x decreases without bound.