(4) Let f(x) be the function with domain R given by the rule X if x EQ, and 0 if x # Q. f(x) = = (a) Prove that lime-o f(x) = 0. (b) Prove that if a 0, then limx→a f(x) does not exist. DEFINITION: Let f be a function and a E R. We say that the limit of f(x) as x approaches a from the right is L provided: For every > 0, there is some 8 >0 such that for all x satisfying a < x
(4) Let f(x) be the function with domain R given by the rule X if x EQ, and 0 if x # Q. f(x) = = (a) Prove that lime-o f(x) = 0. (b) Prove that if a 0, then limx→a f(x) does not exist. DEFINITION: Let f be a function and a E R. We say that the limit of f(x) as x approaches a from the right is L provided: For every > 0, there is some 8 >0 such that for all x satisfying a < x
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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Transcribed Image Text:(4) Let f(x) be the function with domain R given by the rule
if x = Q, and
if x # Q.
f(x) =
X
0
(a) Prove that limo f(x) = 0.
(b) Prove that if a ‡ 0, then limx→a f(x) does not exist.
DEFINITION: Let f be a function and a € R. We say that the limit of f(x) as x approaches a from
the right is L provided:
For every > 0, there is some d > 0 such that for all x satisfying a < x < a+d, we
have that f is defined at x and also that f(x) − L| < €.
In this case, we write
lim f(x) = L.
x→a+
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