(b) Show by example that 15, Let f: R-R be defined by setting f(x) :=x if x is rational, and f(x) = 0 if x is irrational, (a) Show that f has a limit at x 0. (b) Use a sequential argument to show that if c0, thenf does not have a limit at c. %3D If f ie the restriction of f to I. show
(b) Show by example that 15, Let f: R-R be defined by setting f(x) :=x if x is rational, and f(x) = 0 if x is irrational, (a) Show that f has a limit at x 0. (b) Use a sequential argument to show that if c0, thenf does not have a limit at c. %3D If f ie the restriction of f to I. show
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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Question number (15) only.
This material is Real analysis .
![14. Let c ER and let f: R-R be such that lim ((x))
XC
(a) Show that if L 0, then limf(x) = 0.
XC
(b) Show by example that if L 0, then f may not have a limit at c.
15, Let f: R-R be defined by setting f(x):
(a) Show that f has a limit at x = 0.
(b) Use a sequential argument to show that if c0, then f does not have a limit at c.
16. Letf: R-R, let I be an open interval in R, and let c E I. If f¡ is the restriction of f to I, show
that fi has a limit at c if and only if f has a limit at c, and that the limits are equal.
:= x if x is rational, and f(x) =0 if x is irrational.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8881e8ac-0126-47a1-a10c-6241cc2ee256%2F0c649806-f619-490e-857b-2207b3ea22bf%2F3falajh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:14. Let c ER and let f: R-R be such that lim ((x))
XC
(a) Show that if L 0, then limf(x) = 0.
XC
(b) Show by example that if L 0, then f may not have a limit at c.
15, Let f: R-R be defined by setting f(x):
(a) Show that f has a limit at x = 0.
(b) Use a sequential argument to show that if c0, then f does not have a limit at c.
16. Letf: R-R, let I be an open interval in R, and let c E I. If f¡ is the restriction of f to I, show
that fi has a limit at c if and only if f has a limit at c, and that the limits are equal.
:= x if x is rational, and f(x) =0 if x is irrational.
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