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Theorems Regarding Limits of Functions

We observe that every real sequence $(x_n)_{n\in\mathbb N}$ can be interpreted as a function $f:\mathbb N\to\mathbb R.$ With this observation in mind, all theorems we can prove about the limits of functions will also apply to the limits of sequences. In this section, we present some of these theorems.

| | | | created: 2020-07-10 18:30:08 | modified: 2020-07-10 19:47:56 | by: bookofproofs | references: [581], [8311]

1.Proposition: Limit of a Function is Unique If It Exists

2.Proposition: Arithmetic of Functions with Limits - Sums

3.Proposition: Arithmetic of Functions with Limits - Difference

4.Proposition: Arithmetic of Functions with Limits - Product

5.Proposition: Arithmetic of Functions with Limits - Division

6.Proposition: Preservation of Inequalities for Limits of Functions

7.Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity

8.Theorem: Squeezing Theorem for Functions

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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