# Limits Involving Infinity

## Contents

## Limits involving infinity

### Limits at infinity

Let , and .

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

means that for all , there exists *c* such that
whenever *x* > *c*. Or, symbolically:

- .

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

means that for all there exists *c* such that whenever *x* < *c*. Or, symbolically:

- .

For example,

### 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 , and . The statement **the limit of f as x approaches a is infinity**, denoted

means that for all there exists such that whenever

These ideas can be combined in a natural way to produce definitions for different combinations, such as

For example,

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, does not possess a central limit (which is normal):

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit *does* exist in that context:

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 , namely, it is convenient for 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.

## 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