Limits Which Are Infinite

What is

	3x² + 1
lim		?
x --> 0	x²

As x gets close to zero, the values of (3x² + 1)/x² get larger and larger. There is certainly no real number L which can be the limit. Not only can we not make (3x² + 1)/x² arbitrarily close to some L by choosing x sufficiently close to zero (see the definition of limit), but as x gets close to zero (3x² + 1)/x² grows arbitrarily far away from any given L.

We could simply say that the limit does not exist, but Theseus taught us not to give up so easily!

The solution is to say that the limit is "infinite". This is written as

	3x² + 1
lim		=
x --> 0	x²

What it means is that (3x² + 1)/x² can be made arbitrarily large by choosing any x sufficiently close to zero. In general, we define

Definition of an Infinite Limit

The

lim	f(x)	=	.
x --> a

if and only if f(x) can be made arbitrarily large by choosing any x sufficiently close to (but not equal to) a.

Note: As for a finite limit there must be some d>0 such that f(x) is defined for every x with 0<|x- a|<d.

Variations on Infinite Limits

If our goal is to describe vertical asymptotes as infinite limits, we have more work to do. For example, what is

	1
lim		?
x --> 0	sqrt(x)

Since sqrt(x) is not defined for any number to the left of zero, this limit does not exist:

	1
lim		= DNE.
x --> 0	sqrt(x)

But a RIGHT-HAND limit exists. It makes sense to write

	1
lim		= .
x --> 0⁺	sqrt(x)

Along the same lines, what about a function like f(t) = -1/t² whose values grow negatively large as t approaches 0? To handle this we introduce the symbol - and write

	1
lim		= -.
t --> 0	t²

Graph of v = (u² + 1)/(u² - 2).

Left- and Right-hand limits can be combined with the symbols and - in a number of ways, each of which corresponds to a type of vertical asymptote. To illustrate, the asymptotes of the graph of (u² + 1)/(u² - 2) at x=sqrt(2) can be expressed using limits as follows:

	u² + 1				u² + 1
lim		= .	and	lim		= -.
u --> sqrt(2)⁺	u² - 2			u --> sqrt(2)^-	u² - 2

Note that under any definition

	u² + 1
lim		= DNE
u --> sqrt(2)	u² - 2

since the one-sided limits do not agree.

There is nothing very tricky about infinite limits, especially if we remember two things:

An infinite limit of any sort (one-sided, regular, with a "value" of or -) corresponds to a type of vertical asymptote.
The symbols and - are symbols. They are not numbers.

The last order of business in the Stage is to find limit statements which treat horizontal asymptotes. ONLY ONE MORE PAGE TO GO!

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