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Examples of Limit Calculations

In this section, we will actually calculate some limits for specific functions $f(x)$ as $x$ tends either to a specific real number $a\in\mathbb R$ or to infinity $\infty$. The examples listed below are important since they occur very often as lemmata for other mathematical theorems.

| | | | created: 2020-07-10 18:36:16 | modified: 2020-07-10 18:40:21 | by: bookofproofs | references: [581], [8311]

1.Proposition: Limit of 1/n

2.Proposition: Limit of the Constant Function

3.Proposition: Limit of the Identity Function

4.Proposition: Limit of Nth Powers

5.Proposition: Convergence Behavior of the Sequence \((b^n)\)

6.Proposition: Limit of a Polynomial

7.Proposition: Limit of a Rational Function

8.Proposition: Limit of Nth Root of a Positive Constant

9.Proposition: Limit of Nth Root of N

10.Proposition: Limits of General Powers

11.Proposition: Square Roots

12.Proposition: Limit of Exponential Growth as Compared to Polynomial Growth

13.Proposition: Limits of Logarithm in $[0,+\infty]$

14.Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth

15.Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function

16.Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

[8311] Modler, F.; Kreh, M.: “Tutorium Analysis 1 und Lineare Algebra 1”, Springer Spektrum, 2018, 4. Auflage