Real Function Limits:Infinite

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.

Limits at Infinity

Another kind of limit involves looking at what happens to $f(x)$ as $x$ gets very big. For example, consider the function $f(x)={\frac {1}{x}}$ . As $x$ gets very big, ${\frac {1}{x}}$ gets very small. In fact, ${\frac {1}{x}}$ gets closer and closer to 0 the bigger $x$ gets. Without limits it is very difficult to talk about this fact, because $x$ can keep getting bigger and bigger and ${\frac {1}{x}}$ never actually gets to 0; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does $x$ have to be to be sure that $f(x)$ is really going towards 0?

In this case, we want to say that, however close we want $f(x)$ to get to 0, for $x$ big enough $f(x)$ is guaranteed to get that close. So we have yet another definition.

Definition: (Definition of a limit at infinity)

We call $L$ the limit of $f(x)$ as $x$ approaches infinity if $f(x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently large.

When this holds we write

$\lim _{x\to \infty }f(x)=L$

or

$f(x)\to L\quad {\mbox{as}}\quad x\to \infty$

Similarly, we call $L$ the limit of $f(x)$ as $x$ approaches negative infinity if $f(x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently negative.

When this holds we write

$\lim _{x\to -\infty }f(x)=L$

or

$f(x)\to L\quad {\mbox{as}}\quad x\to -\infty$

So, in this case, we write:

$\quad \lim _{x\to \infty }{\frac {1}{x}}=0$

and say "The limit, as $x$ approaches infinity, equals $0$ ," or "as $x$ approaches infinity, the function approaches 0.

We can also write:

$\lim _{x\to -\infty }{\frac {1}{x}}=0$

because making $x$ very negative also forces ${\frac {1}{x}}$ to be close to $0$ .

Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.

Limits at Infinity of Rational Functions

One special case that comes up frequently is when we want to find the limit at $\infty$ (or $-\infty$) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example, $f(x)={\frac {x^{3}+x-6}{x^{2}-4x+3}}$ is a rational function. Also, any polynomial is a rational function, since $1$ is just a (very simple) polynomial, so we can write the function $f(x)=x^{2}-3$ as $f(x)={\frac {x^{2}-3}{1}}$ , the quotient of two polynomials.

Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.

There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.

• If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both $\infty$ and $-\infty$) is the ratio of the coefficients of the highest terms.

• If the numerator has the highest term, then the fraction is called "top-heavy". If, when you divide the numerator by the denominator the resulting exponent on the variable is even, then the limit (at both $\infty$ and $-\infty$) is $\infty$ . If it is odd, then the limit at $\infty$ is $\infty$ , and the limit at $-\infty$ is $-\infty$ .

• If the denominator has the highest term, then the fraction is called "bottom-heavy" and the limit at both $\pm \infty$ is 0.

Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as $x^{0}$ . Also, a straight power of $x$ , like $x^{3}$ , has coefficient 1, since it is the same as $1x^{3}$ .

Examples

Example 1

Find $\lim _{x\to \infty }{\frac {x-5}{x-3}}$ .

The function $f(x)={\frac {x-5}{x-3}}$ is the quotient of two polynomials, $x-5$ and $x-3$ . By our rule we look for the term with highest exponent in the numerator; it's $x$ . The term with highest exponent in the denominator is also $x$ . So, the limit is the ratio of their coefficients. Since $x=1x$, both coefficients are 1, $\lim _{x\to \infty }{\frac {x-5}{x-3}}={\frac {1}{1}}=1$ .

Example 2

Find $\lim _{x\to \infty }{\frac {x^{3}+x-6}{x^{2}-4x+3}}$.

Using L'Hôpital's rule $d/dx(x^{3}+x-6)=3x^{2}+1$

and

$d/dx(x^{2}-4x+3)=2x-4$

We look at the terms with the highest exponents; for the numerator, it is $x^{3}$ , while for the denominator it is $x^{2}$ . Since the exponent on the numerator is higher, we know the limit at $\infty$ will be $\infty$ . So,

$\lim _{x\to \infty }{\frac {x^{3}+x-6}{x^{2}-4x+3}}=\infty$ .

Formal Definition of Infinite Limits

A graphical example of a function converging to a limit as it approaches infinity

It is very common, given limits, to work with the concept of infinity. However, the concept of infinity has yet to be well defined. Intuitively, we know that infinity represents endlessness and it is represented as ∞. Yet, infinity itself is not a number. The current limit definition will fail if we use infinity like a number. If you suppose some limit where c = ∞ and we use our original definition, it would mean that

$\lim _{x\rightarrow \infty }f(x)=L$  means that  $\forall M>0:\exists \delta :0<|x-\infty |<\delta \implies |f(x)-L|<\epsilon$

Which is clearly nonsense!

1. You cannot "subtract by infinity" - infinity isn't a number nor is it really a variable.
2. Infinity cannot be bounded, yet by putting infinity in a $|a-b|<x$ format, it implies boundedness.

So, the definition needs to be rewritten, which is done in the following chart. The definitions for when either the limit as x approaches positive or negative infinity; or the limit as ƒ(x) converges to positive or negative infinity are as follows:

Note: Yes, the approaching and converging distinction is important. You can look at it as either referencing the delta or the epsilon, respectively.}}

Variations of the Epsilon-Delta Definition

Notation	Formulation
$\lim _{x\rightarrow c}f(x)=\infty$	$\forall N>0:\exists \delta :0<|x-c|<\delta \implies f(x)>N$
$\lim _{x\rightarrow c}f(x)=-\infty$	$\forall N>0:\exists \delta :0<|x-c|<\delta \implies f(x)<-N$
$\lim _{x\rightarrow \infty }f(x)=L$	$\forall \epsilon >0:\exists M:x>M\implies |f(x)-L|<\epsilon$
$\lim _{x\rightarrow -\infty }f(x)=L$	$\forall \epsilon >0:\exists M:x<-M\implies |f(x)-L|<\epsilon$
$\lim _{x\rightarrow \infty }f(x)=\infty$	$\forall N>0:\exists M:x>M\implies f(x)>N$
$\lim _{x\rightarrow \infty }f(x)=-\infty$	$\forall N>0:\exists M:x>M\implies f(x)<-N$
$\lim _{x\rightarrow -\infty }f(x)=\infty$	$\forall N>0:\exists M:x<-M\implies f(x)>N$
$\lim _{x\rightarrow -\infty }f(x)=-\infty$	$\forall N>0:\exists M:x<-M\implies f(x)<-N$

Take a note of the following variables:

1. N usually notates a limit with infinity and is analogous to ε.
2. M usually notates a limit with infinity and is analogous to δ.

We only use big N and M because the connotation associated with ε and δ is that they are small numbers. Big N and M has the opposite connotation.

Resources

• Infinite limits, Wikibooks: Calculus
• Limits, Wikibooks: Real Analysis
• Limit of a function, Wikipedia