Question 1 Describe all limits points of A where (a) A = ZU (0, 1), (b) A = {1/n : n € N}.
Question 1 Describe all limits points of A where (a) A = ZU (0, 1), (b) A = {1/n : n € N}.
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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MTHS311_Tut...
MTHS311 - Real Analysis
Tutorial 3
Question 1
Describe all limits points of A where
(a) A = ZU (0, 1),
(b) A = {1/n : n e N}.
Question 2
Use only the e – ô definition of a limit to show that:
(a) lim-2 x2 = 4,
(b) lim-2 A = 4,
(c) lim,0 x sin ! = 0,
(d) lim,→0
= 0.
Question 3
By using the Rules for limits evaluate
(a) lim,→1 을 (x > 0),
(b) lim,-0 (x € R),
(c) lim→-2 212 +7x+6'
1²++x-2
(d) lim 2+7a+6*
Question 4
Consider the function f : R → {0, 1} given by
(1 if r € Q
f(x) =
10 if r ER\Q.
1
![Mathematics
MTHS311 - Real Analysis
Page 2 of 3
Show that if a E R, then lim, a f(x) does not exist.
Question 5
Consider the function
r -1 if r € R is rational
f(r) =
5 - z if r € R is irrational.
Show that lim,-3 f(x) exists but lim,ta does not exist for any a +3.
Question 6
Evaluate the following limits, or show that they do not exist
(a) lim, 1+ (r # 1),
(b) lim,15 (x # 1),
(c) lim, 0+(x + 2)/VE (x > 0),
(d) lim, (r + 2)/VE (1 > 0),
(e) lim, 0(V+1)/r (x > -1),
(f) lim, (VI +I)/x (r > 0).
Question 7
Use the definition of continuity at a point to prove that
(a) f(x) = 3x – 5 is continuous at r = 2.
(b) f(x) = x? is continuous at r = 3.
(c) f(r) = 1/r is continuous at r = 1/2.
Question 8
Give a complete discussion of the continuity properties (including type of discontinuities)
of the functions below in the indicated domain. Prove your answer using definitions only.
(a) f(x) = r*, r €R.
(b) f(x) = x², x € (3,0) U (0, 2).
(c)
1 if r€ [0,1]
f(x) =
if r € (1,5] .
Please go on to the next page...
Mathematics
MTHS311 - Real Analysis
Page 3 of 3
Question 9
Determine which of the following functions are uniformly continuous:
(a) f(x) = Inr on (0, 1).
(b) f(r) = x sin z on (0, 1).
(c) f(x) = Va on (0, 0).
(d) f(x) = e on (0, 0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49588333-9bdd-4034-afa3-cea7dcd808af%2Fee106e5c-eee4-4971-a486-b0d6e0f6abc7%2F31yz9tj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Mathematics
MTHS311 - Real Analysis
Page 2 of 3
Show that if a E R, then lim, a f(x) does not exist.
Question 5
Consider the function
r -1 if r € R is rational
f(r) =
5 - z if r € R is irrational.
Show that lim,-3 f(x) exists but lim,ta does not exist for any a +3.
Question 6
Evaluate the following limits, or show that they do not exist
(a) lim, 1+ (r # 1),
(b) lim,15 (x # 1),
(c) lim, 0+(x + 2)/VE (x > 0),
(d) lim, (r + 2)/VE (1 > 0),
(e) lim, 0(V+1)/r (x > -1),
(f) lim, (VI +I)/x (r > 0).
Question 7
Use the definition of continuity at a point to prove that
(a) f(x) = 3x – 5 is continuous at r = 2.
(b) f(x) = x? is continuous at r = 3.
(c) f(r) = 1/r is continuous at r = 1/2.
Question 8
Give a complete discussion of the continuity properties (including type of discontinuities)
of the functions below in the indicated domain. Prove your answer using definitions only.
(a) f(x) = r*, r €R.
(b) f(x) = x², x € (3,0) U (0, 2).
(c)
1 if r€ [0,1]
f(x) =
if r € (1,5] .
Please go on to the next page...
Mathematics
MTHS311 - Real Analysis
Page 3 of 3
Question 9
Determine which of the following functions are uniformly continuous:
(a) f(x) = Inr on (0, 1).
(b) f(r) = x sin z on (0, 1).
(c) f(x) = Va on (0, 0).
(d) f(x) = e on (0, 0).
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