Limits Involving Infinity

Limits involving infinity

Limits at infinity

The limit of this function at infinity exists.

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:

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

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 a ∈ R is defined in the normal way metric space R.

In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ 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

Horizontal asymptote about y = 4

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.

Resources

• Limits at Infinity Part I, Paul's Online Notes
• Limits at Infinity Part II, Paul's Online Notes
• Introduction to limits at infinity, Khan Academy

Licensing

Content obtained and/or adapted from:

• Limit of a function, Wikipedia under a CC BY-SA license