Infinity or Undefined?

Date: 06/21/97 at 02:30:41
From: Luke Hawkins
Subject: Tan90=infinity, doesn't it?

Dear Dr. Math,
	
I have read your explanation of why tan(90) is undefined. Wouldn't it 
be infinity? I can always divide nothing into something. Also, on the 
tangent graph, when you reach 90 and 270, the point is plotted at 
infinity. 

Thanks,
Luke Hawkins

Date: 06/22/97 at 15:43:30
From: Doctor Mike
Subject: Re: Tan90=infinity, doesn't it?

Hi Luke,
   
I understand why you would think this. But here is the reason why we 
say "undefined" instead of "equals infinity." Each of the trig 
functions is a "real function."  A real function gives a way (a rule) 
to start out with a REAL NUMBER and wind up with another REAL NUMBER.  
An easy example: 
   
        F(x) = 4*x
  
This is the "four times" function. For every real number x you can 
easily find the number 4*x. It always makes sense to talk about "the 
number that is four times another number," right? The trouble with 
the function:  
   
        T(x) = tan(x)
   
is that "the number that is the tangent of x" does not always make 
sense. This is because "infinity" is not a real number, it is more of 
an idea that is "way out there beyond any real numbers." You can't do 
things with infinity the way you can with real numbers. You cannot 
subtract 10 from it and get something smaller. You cannot multiply it 
by zero and get zero, etc.  
   
So, the tangent of a real number angle is a very useful function for 
most angles, but it doesn't make sense for all of them. For those 
angles, like 90 degrees, we just say as much as we can, namely that as 
the angle gets closer and closer to 90 degrees, the tangent of that 
sequence of angles gets bigger and bigger in absolute value. Notice 
that I cannot just say "bigger and bigger". A much different thing 
happens for the tangents of the angles that are just a bit smaller 
than 90, versus the angles just a bit larger. Try it. Find the 
tangents of:
 
     89   89.9   89.99   89.999   89.9999  89.99999    ....

Then find the tangents of:

     91   90.1   90.01   90.001   90.0001  90.00001   ...
   
I hope this helps you to understand the terminology.

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