growth of exponential function

Lemma.

$$\underset{x\to \mathrm{\infty }}{lim}\frac{x^a}{e^x}=0$$

for all values of $a$.

Proof.  Let $\epsilon$ be any positive number.  Then we get:

as soon as  $x>\max \{1,\frac{(\lceil a\rceil +1)!}{\epsilon }\}$.  Here, $\lceil \cdot \rceil$ the ceiling function;  $e^x$ has been estimated downwards by taking only one of the all positive

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\mathrm{\cdots }+\frac{x^n}{n!}+\mathrm{\cdots }$$

Theorem.

The of the real exponential function   $x\mapsto b^x$   exceeds all power functions, i.e.

$$\underset{x\to \mathrm{\infty }}{lim}\frac{x^a}{b^x}=0$$

with $a$ and $b$ any ,  $b>1$.

Proof.  Since  $\ln b>0$,  we obtain by using the lemma the result

$$\underset{x\to \mathrm{\infty }}{lim}\frac{x^a}{b^x}=\underset{x\to \mathrm{\infty }}{lim}{\left(\frac{x^{\frac{a}{\ln b}}}{e^x}\right)}^{\ln b}=0^{\ln b}=0.$$

Corollary 1.  $\underset{x\to 0+}{lim}x\ln x=0.$

Proof.  According to the lemma we get

$$0=\underset{u\to \mathrm{\infty }}{lim}\frac{-u}{e^u}=\underset{x\to 0+}{lim}\frac{-\ln \frac{1}{x}}{\frac{1}{x}}=\underset{x\to 0+}{lim}x\ln x.$$

Corollary 2.  $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{\ln x}{x}}=0.$

Proof.  Change in the lemma  $x$  to  $\ln x$.

Corollary 3.  $\underset{x\to \mathrm{\infty }}{lim}{x}^{\frac{1}{x}}=1.$   (Cf. limit of nth root of n.)

Proof.  By corollary 2, we can write:  $x^{\frac{1}{x}}=e^{\frac{\ln x}{x}}⟶e^0=1$  as  $x\to \mathrm{\infty }$ (see also theorem 2 in limit rules of functions).

Title	growth of exponential function
Canonical name	GrowthOfExponentialFunction
Date of creation	2013-03-22 14:51:32
Last modified on	2013-03-22 14:51:32
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	18
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26A12
Classification	msc 26A06
Related topic	MaximalNumber
Related topic	LimitRulesOfFunctions
Related topic	NaturalLogarithm
Related topic	AsymptoticBoundsForFactorial
Related topic	MinimalAndMaximalNumber
Related topic	FunctionXx
Related topic	Growth
Related topic	LimitsOfNaturalLogarithm
Related topic	DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives