# growth of exponential function

Lemma.

 $\lim_{x\to\infty}\frac{x^{a}}{e^{x}}=0$

for all values of $a$.

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

 $0<\frac{x^{a}}{e^{x}}\leqq\frac{x^{\lceil a\rceil}}{e^{x}}<\frac{x^{\lceil a% \rceil}}{\frac{x^{\lceil a\rceil+1}}{(\lceil a\rceil+1)!}}=\frac{(\lceil a% \rceil+1)!}{x}<\varepsilon$

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

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

###### Theorem.

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

 $\lim_{x\to\infty}\frac{x^{a}}{b^{x}}=0$

with $a$ and $b$ any ,  $b>1$.

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

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

Corollary 1.$\displaystyle\lim_{x\to 0+}x\ln{x}=0.$

Proof.  According to the lemma we get

 $0=\lim_{u\to\infty}\frac{-u}{e^{u}}=\lim_{x\to 0+}\frac{-\ln{\frac{1}{x}}}{% \frac{1}{x}}=\lim_{x\to 0+}x\ln{x}.$

Corollary 2.$\displaystyle\lim_{x\to\infty}\frac{\ln{x}}{x}=0.$

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

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

Proof.  By corollary 2, we can write:  $\displaystyle x^{\frac{1}{x}}=e^{\frac{\ln{x}}{x}}\longrightarrow e^{0}=1$  as  $x\to\infty$ (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