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.