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Zero Laws and L'Hopital's Rule

Date: 03/04/98 at 06:51:19
From: Sheryl
Subject: Zero Laws

Hi, I was just wondering - if you have 0/0 (zero divided by zero), 
which law takes precedence - a) zero divided by any number is zero, 
or b) any number divided by zero is undefined, or c) any number 
divided by itself is one?  Thanks.

Date: 04/02/98 at 13:46:08
From: Doctor Sonya
Subject: Re: Zero Laws

Hi there, Sheryl. 

Zero is a tricky and subtle beast - it does not comform to the usual 
laws of algebra as we know them.  

You are right that zero divided by any number (except zero itself) is 
zero. Put more mathematically:

     0/n = 0      for all non-zero numbers n.

You get into the tricky realms when you try to divide by zero itself.  
It's not true that a number divided by 0 is always undefined. It 
depends on the problem. I'm going to give you an example from calculus 
where the number 0/0 is defined. If you haven't had calculus yet, just 
let this sit in the back of your head, and refer to it again later.

Say we have two functions, f(x) and g(x). If you take their quotient, 
f(x)/g(x), you get another function.  Let's call this h(x). 

You can also take limits of functions. For example, the limit of a 
function f(x) as x goes to 2 is the value that the function gets 
closest to as it takes on x's that approach 2. We would write this 
limit as:

     lim{x->2} f(x) 

This is a pretty intuitive notion. Just draw a graph of your function, 
and move your pencil along it. As the x values approach 2, see where 
the function goes.

Now for our example. Let:

     f(x) = 2x - 2
     g(x) = x - 1

and consider their quotient:
     h(x) = f(x)/g(x)

What if we want the lim{x->1} h(x)? There are theorems that say that

     lim{x->1} h(x) = 
     lim{x->1} f(x)/g(x) =
     (lim{x->1} f(x))/(lim{x->1} g(x)) = 
     0 / 0 !!!

So we now have:
     lim{x->1} h(x) = 0/0

But I can employ another theorem, called l'Hopital's rule, that tells 
me that this limit is also equal to 2. So in this case, 0/0 = 2 
(didn't I tell you it was a strange beast?)

Here's another bit of weirdness with 0. Let's say that 0/0 followed 
that old algebraic rule that anything divided by itself is 1.  Then 
you can do the following proof:

We're given that:

     0/0 = 1

Now multiply both sides by any number n.

     n * (0/0) = n * 1

Simplify both sides:

     (n*0)/0 = n 
     (0/0) = n 

Again, use the assumption that 0/0 = 1:

     1 = n 

So we just proved that all other numbers n are equal to 1!  So 0/0 
can't be equal to 1.

For more answers in our archives about dividing by zero, see the Dr. 
Math FAQ:   

I hope this clears up some of your questions.  I promise, as you study 
more math, zero will continue to surprise you.

-Doctors Sonya and Celko,  The Math Forum
Check out our web site!   
Associated Topics:
High School Calculus
High School Number Theory

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