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Deteraitne wbetber each ef tbe stateneTis below are 'Tras and Fabs. If tbe statEueTi is 'Ins, rove it. I xet, rovids a comder srampls. Es SIFE to jestily wby your conisreYampe vorka, Ebongh a full preef is HuusoDSSaary. (2) IF g: (0, c); –Ris a ceotiuroEs hnctien snch that lms(z; = 0, tben b fg: (0,c0) →Bis au itsgrabls froction, and iben fm gr}=1. eHg:(0,00) →Ris abscltsly sergent, and h : (0, cn) →R is a bonnded frastien, iben lm g(z} = , then sh(z>dz cc0vergea.

Deteraitne wbetber each ef tbe stateneTis below are 'Tras and Fabs. If tbe statEueTi is 'Ins, rove it. I xet, rovids a comder srampls. Es SIFE to jestily wby your conisreYampe vorka, Ebongh a full preef is HuusoDSSaary. (2) IF g: (0, c); –Ris a ceotiuroEs hnctien snch that lms(z; = 0, tben b fg: (0,c0) →Bis au itsgrabls froction, and iben fm gr}=1. eHg:(0,00) →Ris abscltsly sergent, and h : (0, cn) →R is a bonnded frastien, iben lm g(z} = , then sh(z>dz cc0vergea.

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 statements below are True and False. If the statement is True, prove it. If not, provide a counter example. Be sure to justify why your counterexample works, though a full proof is unneccessary. (a) If g : (0, 00) → R is a continuous function such that lim g(x) = 0, then | g(x)dr converges. (b) If g : (0, 00) →R is an integrable function, and converges, then lim g(x) = 0. (c) If g : (0, 0) → R is absolutely convergent, and h : (0, 00) → R is a bounded function, then lim g(x) = 0, then | g(x)h(x)dx converges.
Transcribed Image Text:Determine whether each of the statements below are True and False. If the statement is True, prove it. If not, provide a counter example. Be sure to justify why your counterexample works, though a full proof is unneccessary. (a) If g : (0, 00) → R is a continuous function such that lim g(x) = 0, then | g(x)dr converges. (b) If g : (0, 00) →R is an integrable function, and converges, then lim g(x) = 0. (c) If g : (0, 0) → R is absolutely convergent, and h : (0, 00) → R is a bounded function, then lim g(x) = 0, then | g(x)h(x)dx converges.
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