H 00:51 ()0.00KB/s61%MTHS311_Tut...MTHS311 - Real AnalysisTutorial 3Question 1Describe all limits points of A where(a) A = ZU (0, 1),(b) A = {1/n : n e N}.Question 2Use 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 3By 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 4Consider the function f : R → {0, 1} given by(1 if r € Qf(x) =10 if r ER\Q.1 MathematicsMTHS311 - Real AnalysisPage 2 of 3Show that if a E R, then lim, a f(x) does not exist.Question 5Consider the functionr -1 if r € R is rationalf(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 6Evaluate 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 7Use 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 8Give 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...MathematicsMTHS311 - Real AnalysisPage 3 of 3Question 9Determine 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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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

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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.

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