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growth of exponential function (Topic)

Lemma.

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

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

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

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

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

$\displaystyle \lim_{x\to\infty}\frac{x^a}{b^x} = 0$
with $a$ and $b$ any constants,  $b > 1$.

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

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

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

Proof.  According to the lemma we get

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

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).



"growth of exponential function" is owned by pahio. [ full author list (2) ]

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Also defines:  ceiling function, real exponential function, power function

Cross-references: functions, theorem
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This is version 4 of growth of exponential function, born on 2009-04-17, modified 2009-04-18.
Object id is 645, canonical name is GrowthOfExponentialFunction.
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Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
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