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Section: 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.

Table of Contents

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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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition