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Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function. Suppose that c e R and that for some n > 0, (c-7, c) U (c, c+n) K. The limit of f as r approaches c exists if and only if the limit of f as x approaches c from the right and the limit of f as x approaches e from the left both exist and are equal.

Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function. Suppose that c e R and that for some n > 0, (c-7, c) U (c, c+n) K. The limit of f as r approaches c exists if and only if the limit of f as x approaches c from the right and the limit of f as x approaches e from the left both exist and are equal.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 5.2.3 is the crucial link between one-sided and two-sided limits. Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function. Suppose that c e R and that for some n > 0, (c- 7, c) U (c, c+ 7) C K. The limit of f as r approaches c exists if and only if the limit of f as x approaches c from the right and the limit of f as z approaches c from the left both exist and are equal. Problems 5.2 lo im 1. Prove Theorem 5.2.3, which establishes the relationship between one- and two-sided limits. P is a monotonic function provided 1. f is monotonic
Transcribed Image Text:Theorem 5.2.3 is the crucial link between one-sided and two-sided limits. Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function. Suppose that c e R and that for some n > 0, (c- 7, c) U (c, c+ 7) C K. The limit of f as r approaches c exists if and only if the limit of f as x approaches c from the right and the limit of f as z approaches c from the left both exist and are equal. Problems 5.2 lo im 1. Prove Theorem 5.2.3, which establishes the relationship between one- and two-sided limits. P is a monotonic function provided 1. f is monotonic
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