Approaching Zero and Losing the Plot

Date: 11/11/2010 at 21:38:35
From: Andrew
Subject: why doesn't y=x^0 =0?

The smaller the exponent of x in functions of the type y = x^n, the closer
the graphs tend to the x-axis:

                                                           is a line
   y = x^1                                                 through the
                                                           origin
                                                           

                                                           is a parabola
   y = x^0.5                                               about the
                                                           x-axis


   y = x^0.00000000000000000000000000000000000000000001    appears to tend
                                                           towards y = 0

So why doesn't y = x^0 = 0? Why does it jump to y = 1, instead?

Date: 11/11/2010 at 22:16:10
From: Doctor Ali
Subject: Re: why doesn't y=x^0 =0?

Hi Andrew!

Thanks for writing to Dr. Math.
 
There are two points to consider in order to understand this.

First, note that even 
x^0.00000000000000000000000000000000000000000001
goes to infinity as x goes to infinity. It may grow very, very slowly --
but it will still approach infinity.

Second, beware the domain between 0 and 1. As close inspection of your
graphing investigation will reveal, when you raise these values to smaller
and smaller powers, they tend to 1. The values after 1 return y's which
are slightly more than 1 -- but they are still more than 1.

With these two thoughts in mind, you'll see that a function which is made
by raising x to a small power is "mostly around 1," if you will. So no
wonder x^0 equals 1.

Now consider the natural numbers:

   n: 1, 2, 3, 4, ...

Ignoring the negative x's, think about the way y = x^n behaves for x's
between 0 and 1. Plot them all in one system and see what happens as you
raise x to increasingly larger powers.

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/