Limits as the variable tends to infinity

Summary

This lesson studies the behavior of a function as the variable grows without bound, and carefully distinguishes it from another situation that is written with the same symbol: that of a function whose values grow without bound. The limit as the variable tends to infinity is defined with epsilon and a bound $M$ , and it is proved only once that it is equivalent to a one-sided limit at $0$ by means of the change of variable $t = \frac{1}{x}$ . It follows that all the theorems of the preceding lessons also hold at infinity, without a new proof. With them the limits of powers and of rational functions are computed, which proves the horizontal and oblique asymptotes that Algebra and Geometry admitted; the squeeze theorem and the theorem on monotonic functions are applied; and one recognizes when the limit does not exist.

Learning objectives

Distinguish the limit as the variable tends to infinity from the infinite limit, and write both with the common scheme of challenge and response.
Define with epsilon and a bound $M$ the limit as the variable tends to infinity, and prove its equivalence with a one-sided limit at $0$ .
Prove the limits of powers and of rational functions as the variable tends to infinity, and deduce from them the horizontal and oblique asymptotes.
Compute limits at infinity with the squeeze theorem, monotonic functions and changes of variable, and prove when they do not exist.

Two uses of infinity

The word “infinity” appears in the study of limits with two different meanings, which should be separated from the outset. In the first, it is the variable that grows without bound, while the values of the function approach a number. In the second, it is the function that grows without bound, while the variable approaches a number. The three following statements are therefore different.

$\lim_{x \to \infty} f(x) = L$ : the variable grows without bound and the values of $f$ approach the number $L$ . For example, $\frac{1}{x}$ approaches $0$ as $x$ grows. It is the subject of this lesson.
$\lim_{x \to a} f(x) = \infty$ : the variable approaches the number $a$ and the values of $f$ grow without bound. For example, $\frac{1}{x^2}$ grows without bound as $x$ approaches $0$ . It is the subject of the lesson on infinite limits.
$\lim_{x \to \infty} f(x) = \infty$ : both grow without bound, like $x^2$ . It is also studied in the lesson on infinite limits.

The symbol $\infty$ is not a real number, and the two situations are not symmetric. In the first, the limit exists and is a number, $L$ , to which uniqueness, the algebra of limits and all the known theorems apply. In the second, the function has no limit in the sense of the lesson on the limit of functions of a real variable: the notation $= \infty$ does not name a limit, but a precise way of not having one. That is why the rules of the algebra of limits, which operate on numbers, cannot be applied to $\infty$ as if it were one.

Both situations fit, however, into one and the same scheme. Every definition of limit says: for every challenge on the values there exists a response on the variable. The only thing that changes is what it means to be near each thing:

the variable is near the number $a$ if $0 < \abs{x - a} < \delta$ , and near $\infty$ if $x > M$ ;
the value is near the number $L$ if $\abs{f(x) - L} < \varepsilon$ , and near $\infty$ if $f(x) > N$ .

The definition of the lesson on the limit combines the first form of each line; this lesson, the second form of the first line with the first form of the second; the lesson on infinite limits, the remaining ones. The first proposition of this lesson shows the two uses with one and the same function.