Part C) Use the def of e - g to prove that f(x) is continuous at x=0. (a) For f(r) = x² sin(x), find a partition P of [-5, ] such that U(f, P) – L(f, P) < : Notesin() = 1/2.200(b) Use only limit definition of a derivative and the fact thatsin(r)lim= 1,to directly find f'(7) for f(x) = x² sin(x).(c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) iscontinuous at r = 0. Do not assume/use continuity of x² or sin(x).

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Answered: ổ definition of continuous functions to…

ổ definition of continuous functions to directly prove f(x) = x² sin(x) is = 0. Do not assume/use continuity of x² or sin(x). (c) Use only the e - continuous at 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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Part C) Use the def of e - g to prove that f(x) is continuous at x=0.

(a) For f(r) = x² sin(x), find a partition P of [-5, ] such that U(f, P) – L(f, P) < : Note
sin() = 1/2.
200
(b) Use only limit definition of a derivative and the fact that
sin(r)
lim
= 1,
to directly find f'(7) for f(x) = x² sin(x).
(c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) is
continuous at r = 0. Do not assume/use continuity of x² or sin(x).

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