9.2.2 Notation for Limiting Values

Improper integrals bring us — again — to the study of limiting behavior of a function as its independent variable becomes large. In particular, the first step in evaluating

is to find the antiderivative

The improper integral is then the limiting value of `G text[(] T text[)]` as `T rarr oo`. The standard notation for this limiting value is

With this notation we may condense the two-step definition of the improper integral into a single formula:

Notation   The limiting value of a function `f text[(] x text[)]` as the independent variable `x` becomes large is denoted lim x → ∞ f ( x ) .

Example 2

Express in limiting notation the eventual behavior, as the independent variable becomes large, of the function

Solution   We have a numerator approaching `- oo` and a denominator approaching `oo`, so we need to transform the fraction algebraically:

Check this by using your graphing tool to graph `f` for large values of `x`.

Example 3

Express in limiting notation the eventual behavior, as the independent variable becomes large, of the function

Solution   This example is similar to Example 2, but the notation is slightly different, and the outcome is different:

Activity 1

Make a list of previous occurrences in this text of limiting behavior as an independent variable becomes large. Express each item on your list in the new notation.

Comment on Activity 1

Checkpoint 2