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/
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