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Prove the following limits of functions, using only the e-8 Definition. Definition (Functional Limit). Let f: A → R, and let c be a limit point of the domain A. We say that lime f(x) = L provided that, for all € > 0, there exists a > 0 such that whenever 0 < x-cl < 8 (and a € A) it follows that f(x) - L

Prove the following limits of functions, using only the e-8 Definition. Definition (Functional Limit). Let f: A → R, and let c be a limit point of the domain A. We say that lime f(x) = L provided that, for all € > 0, there exists a > 0 such that whenever 0 < x-cl < 8 (and a € A) it follows that f(x) - L

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Prove the following limits of functions, using only the e-d Definition. Definition (Functional Limit). Let f: A → R, and let c be a limit point of the domain A. We say that lima c f(x) = L provided that, for all € > 0, there exists a 8 >0 such that whenever 0 < x- c < 8 (and a € A) it follows that f(x) - L<e. (a) lim 2x + 4 = 8 x 2 (b) lim x³ x-0 (c) lim x³ x-2 = - 0; 8; This is often referred to as the "e & version" of the definition for functional. limits. Recall that the statement. f(x) - L<e is equivalent to f(x) = V(L). Likewise, the statement |xc|< 8 is satisfied if and only if a € Vs (c).
Transcribed Image Text:Prove the following limits of functions, using only the e-d Definition. Definition (Functional Limit). Let f: A → R, and let c be a limit point of the domain A. We say that lima c f(x) = L provided that, for all € > 0, there exists a 8 >0 such that whenever 0 < x- c < 8 (and a € A) it follows that f(x) - L<e. (a) lim 2x + 4 = 8 x 2 (b) lim x³ x-0 (c) lim x³ x-2 = - 0; 8; This is often referred to as the "e & version" of the definition for functional. limits. Recall that the statement. f(x) - L<e is equivalent to f(x) = V(L). Likewise, the statement |xc|< 8 is satisfied if and only if a € Vs (c).
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