proof of growth of exponential function

In this proof, we first restrict to when x and a are integers and only later lift this restricton.

Let a>0 be an integer, let b>1 be real, and let x be an integer.

Consider the following inequality


If x2, then we have


Define X to be the greater of 2 and a(3/2)a-1/(1-b); when x>X, we have


Rewrite xa/bx as follows when x>X:


By the inequality established above, each term in the productMathworldPlanetmathPlanetmath will be bounded by 1/b, hence


Since b>1, it is also the case that b>1, hence we have the inequality


Combining the last two inequalities yields the following:


From this, it follows that limxxa/bx=0 when a and x are integers.

Now we lift the restrictionPlanetmathPlanetmathPlanetmath that a be an integer. Since the power function is increasing, xa/bxxa/bx, so we have limxxa/bx=0 for real values of a as well.

To lift the restriction on x, let us write x=x1+x2 where x1 is an integer and 0x2<1. Then we have


If x>2, then (x1+x2)/x2<1.5. Since x20,b-x21. Hence, for all real x>2, we have


From this inequality, it follows that limxxa/bx=0 for real values of x as well.

Title proof of growth of exponential function
Canonical name ProofOfGrowthOfExponentialFunction
Date of creation 2013-03-22 15:48:36
Last modified on 2013-03-22 15:48:36
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Proof
Classification msc 32A05