PlanetMath: growth of exponential function

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growth of exponential function

(Theorem)

Lemma.

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

for all constant values of

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; 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 1 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

and

any constants,

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 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.

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Attachments:: proof of growth of exponential function (Proof) by rspuzio

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Cross-references: limit rules of functions, limit of nth root of n, power functions, exponential function, real, ceiling function, positive
There are 9 references to this entry.

This is version 13 of growth of exponential function, born on 2004-11-26, modified 2007-12-07.
Object id is 6532, canonical name is GrowthOfExponentialFunction.
Accessed 3311 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	26A12 (Real functions :: Functions of one variable :: Rate of growth of functions, orders of infinity, slowly varying functions)

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