Calculus: Limits

Date: 11/24/97 at 07:52:20
From: Susan Batchilder
Subject: Calculus: limits

In my textbook, the limit of (1/x) as x approaches zero is defined as 
infinity. But the limit as you approach zero from the negative side is 
negative infinity and the limit as you approach zero from the positive 
side is positive infinity. Therefore since the limit from the right 
does not equal the limit from the left, then by definition limit should 
not exist.

So, why does my textbook have the limit defined as infinity?

I look forward to having the mystery solved.

Thank you, 
Susan

Date: 11/24/97 at 12:32:16
From: Doctor Rob
Subject: Re: Calculus:  limits

Dear Susan,

Your reasoning showing that the limit does not exist is correct.  
Because the limit does not exist, saying that it "equals infinity" is, 
on the face of it, nonsensical.  

This function behaves in a special way that makes the limit not exist. 
The way it behaves is that as x gets small in absolute value, the 
function gets large in absolute value.  Stated more technically, given 
any number N > 0, we can find a d > 0 such that whenever |x| < d, then 
|1/x| > N.  

The value of d happens to be 1/N for this particular function, but 
would be different for other functions.

This means that as x gets small in absolute value, the function grows 
without bound in absolute value.  The function is said to have a 
vertical asymptote at x = 0, and the common phrase that the limit 
"equals infinity" is actually a shorthand for this situation.  So this 
is why your textbook might have said that the limit was infinity.

But why is this phrase chosen?  It has to do with convenience.  
Mathematicians like the fact that limits, when they exist, commute 
with all the operations of arithmetic.  We want this property to hold 
true for all limits, and when we try to extend it to include limits of 
functions like 1/x as x -> 0, we end up with exactly the same 
structure as when we try to append "infinity" to the usual real 
numbers, and the same difficulties.

Examples of these difficulties are infinity minus infinity not being 
definable, infinity over infinity not being definable, and infinity 
times zero not being  definable.

Even though working with infinity as a number gives rise to all of the 
difficulties above, it is still easier than working with something 
that's "undefined."

There are other ways that limits might not exist, but do not "equal 
infinity," or have their absolute values get arbitrarily large.  One 
example is the function f(x) defined by:
   
   f(x) = -1 if x < 0 
   f(x) = 0 if x = 0 
   f(x) = +1 if x > 0.  

f(x) is a function whose limit as x approaches zero does not exist, 
because there is a jump at 0.  A more difficult example is the 
function:
   
   sin(1/x) as x -> 0

for which the limit also doesn't exist, due to ever-increasingly rapid 
oscillation.  In neither of these situations do we use the phrase that 
the limit "equals infinity."

I hope this makes sense.  If not, write again.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/