Talk:PlanetPhysics/Growth of Exponential Function

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: growth of exponential function %%% Primary Category Code: 02.30.-f %%% Filename: GrowthOfExponentialFunction.tex %%% Version: 4 %%% Owner: pahio %%% Author(s): pahio, bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\textbf{Lemma.} $$\lim_{x\to\infty}\frac{x^a}{e^x} = 0$$ for all constant values of $a$.

{\em 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, $\lceil\cdot\rceil$ means the ceiling function;\, $e^x$ has been estimated downwards by taking only one of the all positive terms of the series expansion $$e^x = 1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}+\cdots$$\\

\textbf{\htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}.} The growth 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 constants,\, $b > 1$.

{\em Proof.}\, Since\, $\ln b > 0$,\, we obtain by using the lemma the result $$\lim_{x\to\infty}\frac{x^a}{b^x} = \lim_{x\to\infty}\left(\frac{x^{\frac{a}{\ln b}}}{e^x}\right)^{\ln b} = 0^{\ln b} = 0.$$\\

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

{\em Proof.}\, According to the lemma we get $$0 = \lim_{u\to\infty}\frac{-u}{e^u} = \lim_{x\to 0+}\frac{-\ln{\frac{1}{x}}}{\frac{1}{x}} = \lim_{x\to 0+}x\ln{x}.$$\\

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

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

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

{\em 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 \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html}).

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