# Limits Involving Infinity

## Limits involving infinity

### Limits at infinity

Let $S\subseteq \mathbb {R}$ , $x\in S$ and $f:S\mapsto \mathbb {R}$ .

The limit of f as x approaches infinity is L, denoted

$\lim _{x\to \infty }f(x)=L,$ means that for all $\varepsilon >0$ , there exists c such that $|f(x)-L|<\varepsilon$ whenever x > c. Or, symbolically:

$\forall \varepsilon >0\;\exists c\;\forall x>c:\;|f(x)-L|<\varepsilon$ .

Similarly, the limit of f as x approaches negative infinity is L, denoted

$\lim _{x\to -\infty }f(x)=L,$ means that for all $\varepsilon >0$ there exists c such that $|f(x)-L|<\varepsilon$ whenever x < c. Or, symbolically:

$\forall \varepsilon >0\;\exists c\;\forall x .

For example,

$\lim _{x\to -\infty }e^{x}=0.\,$ ### Infinite limits

For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values. Let $S\subseteq \mathbb {R}$ , $x\in S$ and $f:S\mapsto \mathbb {R}$ . The statement the limit of f as x approaches a is infinity, denoted

$\lim _{x\to a}f(x)=\infty ,$ means that for all $N>0$ there exists $\delta >0$ such that $f(x)>N$ whenever $0<|x-a|<\delta .$ These ideas can be combined in a natural way to produce definitions for different combinations, such as

$\lim _{x\to \infty }f(x)=\infty ,\lim _{x\to a^{+}}f(x)=-\infty .$ For example,

$\lim _{x\to 0^{+}}\ln x=-\infty .$ Limits involving infinity are connected with the concept of asymptotes.

These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if

• a neighborhood of −∞ is defined to contain an interval [−∞, c) for some c ∈ R,
• a neighborhood of ∞ is defined to contain an interval (c, ∞] where c ∈ R, and
• a neighborhood of aR is defined in the normal way metric space R.

In this case, R is a topological space and any function of the form fX → Y with XY⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

### Alternative notation

Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as R ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} where a neighborhood of ∞ is a set of the form {x: |x| > c}. The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, $x^{-1}$ does not possess a central limit (which is normal):

$\lim _{x\to 0^{+}}{1 \over x}=+\infty ,\lim _{x\to 0^{-}}{1 \over x}=-\infty .$ In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

$\lim _{x\to 0^{+}}{1 \over x}=\lim _{x\to 0^{-}}{1 \over x}=\lim _{x\to 0}{1 \over x}=\infty .$ In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of $\lim _{x\to 0^{-}}{x^{-1}}=-\infty$ , namely, it is convenient for $\lim _{x\to -\infty }{x^{-1}}=-0$ to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

### Limits at infinity for rational functions

There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):

• If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
• If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
• If the degree of p is less than the degree of q, the limit is 0.

If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.