Limits at Infinity

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.