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Exercise 3.2.1. [Used throughout.] Let m, b,c E R. Using only the definition of limits prove that lim (mx+b) = mc +b.

Exercise 3.2.1. [Used throughout.] Let m, b,c E R. Using only the definition of limits prove that lim (mx+b) = mc +b.

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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3.2.1

Please include a formal proof.

Exercise 3.2.1. [Used throughout.] Let m, b,c E R. Using only the definition of limits, prove that lim (mx+b) — тс + b. X→C
Transcribed Image Text:Exercise 3.2.1. [Used throughout.] Let m, b,c E R. Using only the definition of limits, prove that lim (mx+b) — тс + b. X→C
Definition 3.2.1. Let I CR be an open interval, let c E I, let f: I - {c} → R be a function and let LER. The number L is the limit of f as x goes to c, written lim f(x) = L, %3D if for each ɛ > 0, there is some 8 > 0 such that x E I – {c} and |x – c| < d imply |f(x) – L| < ɛ. If lim f(x) = L, we also say that f converges to L as x goes to c. If f converges to some real number as x goes to c, we say that lim f(x) exists.
Transcribed Image Text:Definition 3.2.1. Let I CR be an open interval, let c E I, let f: I - {c} → R be a function and let LER. The number L is the limit of f as x goes to c, written lim f(x) = L, %3D if for each ɛ > 0, there is some 8 > 0 such that x E I – {c} and |x – c| < d imply |f(x) – L| < ɛ. If lim f(x) = L, we also say that f converges to L as x goes to c. If f converges to some real number as x goes to c, we say that lim f(x) exists.
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