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15 August 2003
Properties of Extreme Limits
by George E. Hrabovsky, President, MAST
A Continuation
Last week we explored the ideas of limits resulting in infinity or 0. Today we will examine the properties of such limits, and the algebraic properties of infinity and 0.
Properties of Infinite Limits
We introduced two theorems last time. This time we will examine 3 corollaries to Theorem 2:
Corollary 1
Corollary 2
Corollary 3
In other words;
![FormBox[RowBox[{L/∞, , =, , RowBox[{0 when L, ≠, 0.}]}], TraditionalForm]](HTMLFiles/index_6.gif){kind=small}
Indeterminate Forms
Our corollaries say nothing about these situations,
These situations are called indeterminate forms. This is because they can take on any value that we choose, but they are not unique.
The Math Challenge from Last Time
The math challenge
was to prove two theorems. The first of these is, 
If
as 
then given any arbitrarily large number 
there must be some 
such that 
if 
. This
implies that 
as 
by the definition of the limit. We can easily reverse the argument. Hence
the theorem is proved.
The second theorem
is, ![Underscript[lim, xx_0] f(x)/g(x) = 0 ⟹ Underscript[lim, xx_0] g(x)/f(x) = ∞ .](HTMLFiles/index_17.gif){kind=small}
We
can see that by theorem 1 from last column (proven above) we have 
as 
. Thus,
as 
. By
theorem 1 again, we also have 
as 
. Thus
the theorem is proved.
The Math Challenge
Can you prove the corollaries 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 infinite limits.
Created by Mathematica (August 13, 2003)