growth of exponential function
Lemma.
for all values of .
Proof. Let be any positive number. Then we get:
as soon as . Here, the ceiling function; has been estimated downwards by taking only one of the all positive
Theorem.
Proof. Since , we obtain by using the lemma the result
Corollary 1.
Proof. According to the lemma we get
Corollary 2.
Proof. Change in the lemma to .
Corollary 3. (Cf. limit of nth root of n.)
Proof. By corollary 2, we can write: as (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 |