1) Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and converges to c (but never equal to c), we have limn → ∞ f(an) = ∞. (That is, prove a version of Relating Sequences to Functions for infinite limits. Make sure you handle both directions of the if and only if!) 2) Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) i. Constant functions. ii. Functions f such that f(x) ≤ π. iii. Functions f that are continuous at the point c = 0. 3). Prove that f(x) = |x + 6| is continuous at all points c in ℝ
1) Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and converges to c (but never equal to c), we have limn → ∞ f(an) = ∞. (That is, prove a version of Relating Sequences to Functions for infinite limits. Make sure you handle both directions of the if and only if!) 2) Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) i. Constant functions. ii. Functions f such that f(x) ≤ π. iii. Functions f that are continuous at the point c = 0. 3). Prove that f(x) = |x + 6| is continuous at all points c in ℝ
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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1) Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and converges to c (but never equal to c), we have limn → ∞ f(an) = ∞.
(That is, prove a version of Relating Sequences to Functions for infinite limits. Make sure you handle both directions of the if and only if!)
2) Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.)
i. Constant functions.
ii. Functions f such that f(x) ≤ π.
iii. Functions f that are continuous at the point c = 0.
3). Prove that f(x) = |x + 6| is continuous at all points c in ℝ
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