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8 lim x-1 x-1 (76 - 9x) such that x ER lim(x² – 1) such that x € R x-3

8 lim x-1 x-1 (76 - 9x) such that x ER lim(x² – 1) such that x € R x-3

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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Determine whether each of the two attached limit functions are continuous at specific points with a formal proof using mathematical induction. There are three definitions from the text for reference. Thank you!

Definition 3.2.1. Let ICR be an open interval, let e E 1, let f: 1– {c} → R be a function and let L E R. The number L is the limit of f as x goes to c, written lim f(x) = L., if for each e > 0, there is some 8 > 0 such that x € /- {c} and |x – c| < 8 imply f(x) – L| < E. If limf(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. A Definition 3.3.1. Let ACR be a set, and let f: A–R be a function. 1. Let c E A. The function f is continuous at c if for each ɛ > 0, there is some 8 > 0 such that x E A and |x– c| < 8 imply |f(x) – f(e) < ɛ. The function f is discontinuous at c if f is not continuous at c; in that case we also say that f has a discontinuity at c. 2. The function f is continuous if it is continuous at every number in A. The function f is discontinuous if it is not continuous. A Definition 3.4.1. Let A CR be a set, and let f: A – R be a function. The function f is uniformly continuous if for each ɛ > 0, there is some 8 > 0 such that x, y € A and |r- y| < 8 imply |f(x) – f(y)| < ɛ.
Transcribed Image Text:Definition 3.2.1. Let ICR be an open interval, let e E 1, let f: 1– {c} → R be a function and let L E R. The number L is the limit of f as x goes to c, written lim f(x) = L., if for each e > 0, there is some 8 > 0 such that x € /- {c} and |x – c| < 8 imply f(x) – L| < E. If limf(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. A Definition 3.3.1. Let ACR be a set, and let f: A–R be a function. 1. Let c E A. The function f is continuous at c if for each ɛ > 0, there is some 8 > 0 such that x E A and |x– c| < 8 imply |f(x) – f(e) < ɛ. The function f is discontinuous at c if f is not continuous at c; in that case we also say that f has a discontinuity at c. 2. The function f is continuous if it is continuous at every number in A. The function f is discontinuous if it is not continuous. A Definition 3.4.1. Let A CR be a set, and let f: A – R be a function. The function f is uniformly continuous if for each ɛ > 0, there is some 8 > 0 such that x, y € A and |r- y| < 8 imply |f(x) – f(y)| < ɛ.
lim x-1 x - 1 (-- 8 9x ) such that x € R lim (x2 – 1) such that x E R x-3
Transcribed Image Text:lim x-1 x - 1 (-- 8 9x ) such that x € R lim (x2 – 1) such that x E R x-3
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