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29 August 2003
Limits at Infinity
by George E. Hrabovsky, President, MAST
A Continuation
Last week we completes our arithmetic of infinity. Today, we explore the question of what happens when an independent variable approaches infinity.
Limits at Infinity
So far we have
been considering limits of definable quantities. It seems natural
to eventually ask the question, "What happens to a function as our
independent variable approaches infinity?"
We begin by assuming
that we have an infinite interval, 
. If,
given any 
,
there exists some arbitrarily large number (classically denoted as 
),
,
such that 
for all 
,
then we can then say that 
approaches 
as 
approaches 
. This
can be written 
as 
,
or,
![Underscript[lim, x +∞] f(x) = L .](HTMLFiles/index_13.gif){kind=small}
This is the definition of a limit at infinity.
Two Theorems
I will now present two theorems that are relevant. This first theorem establishes the fact that any problem involving limits at infinity can be reduced to a problem of limits at zero.
Theorem 1
The function
as 
if and only if there exists another function 
as 
from the right. We also have 
as 
if and only if there exists another function 
as 
from the left.
The second theorem establishes the behavior of sums of limits at infinity.
Theorem 2
If the functions
and 
as 
then 
.
Asymptotes
I now introduce
three definitions that are very important.
If either of these conditions is true,
![Underscript[lim, xx_0 +] f(x) = ∞, Underscript[lim, xx_0 -] f(x) = ∞,](HTMLFiles/index_26.gif){kind=small}
then the line
described by 
is called a vertical asymptote
of 
. If,
on the other hand, one of these conditions is true,
![Underscript[lim, x +∞] f(x) = y_0, Underscript[lim, x -∞] f(x) = y_0,](HTMLFiles/index_29.gif){kind=small}
then the line
described by 
is called a horizontal asymptote
of 
. If
the distance between a point 
and a straight line whose angle of inclination is neither 
nor 
such
that the distance approaches zero as 
,
then the straight line is called an inclined
asymptote of 
.
The Math Challenge from Last Time
The task was to prove the two theorems about addition of infinity. The first was the formula,
If we think about, and you really have to squeeze your head hard to squirt this one out, you will realize that we can write the sum of two functions as such,
![FormBox[RowBox[{f(x) + g(x), , =, , RowBox[{f(x)[1 + g(x)/f(x)], RowBox[{Cell[], .}]}]}], TraditionalForm]](HTMLFiles/index_38.gif){kind=small}
By Corollary
3 from, "Properties
of Extreme Limits," we know that 
as 
. As
a result we have
![FormBox[RowBox[{1 + g(x)/f(x), , 1.}], TraditionalForm]](HTMLFiles/index_41.gif){kind=small}
Then, by Corollary
1 form , "Properties of Extreme Limits," 
as 
.
The second theorem was,
Let's assume
that there is some sufficiently large number, 
,
and we have some 
such that 
for 
. Now,
if we have a deleted neighborhood described by 
where 
and 
have the same sign. Now we have 
implies that 
. Since
is arbitrarily large we see that this implies that 
.
The Math Challenge
Can you prove the theorems of this column?
Math Resources for Limits
Online:
http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node37.html
This is a rather uninspired page, but it is the best one that I could find.
Books:
Richard A. Silverman,
1969, Modern Calculus and Analytic
Geometry, MacMillan Company, New York (Dover Publications has
reprinted this book with corrections in 2002). This has a very nice chapter
on limits that includes a very detailed discussion of limits at infinity.
Created by Mathematica (August 28, 2003)