Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.1: Limits
Problem 62E
See similar textbooks
icon
Related questions
Question

Real Analysis 

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
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS