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(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
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.
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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