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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0faa9f23-27f3-4e93-ace2-b4afe4669e81%2Fc7ae9171-2e7f-4498-9521-73a745b1808d%2Fut6gnlj_processed.gif&w=3840&q=75)
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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