Question 1: Let e be a positive number, n e N*, and r e R. In eachof the following cases, find the value of o for which the given state-ment holds. Clearly show your workings.(a) |x – 2| < 8 implies |r³ – 8| <ɛ(b) |æ – 1| < 8 implies< 1/n31+ 2x

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Answered: Question 1: Let e be a positive number,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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(3b) Consider the open sentences P(x): x? Determine all x E S for which P(x) = Q (x) is a false statement. 4 and Q(x): x> 0 over the domain S = R.
(b) Prove the following statements: i. {r €R: x - 4| = 2|a| – 2} = {-6,2} ii. {x ER: |x – 4| < x} = (2, 00) iii. P(AN B) = P(A) N P(B). iv. P(A)U P(B) C P(AU B) 11. %3D || ch DELL P
(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state thevalue of and δ (or M or N and δ) in each case.(i) How close to 4 do we need to take x so that (x/2− 2) < 0.001 ? (ii) How close to 4 do we need to take x so that (x/2− 2) > 0.0001 ?(iii) How close to 0 do we need to take x so that (2x + 9) < 9.0001 ?(iv) How close to 0 do we need to take x so that (2x + 9) > 8.999?(v) How close to 0 do we need to take x so that (x^2 + 6x + 9) < 9.001 ? (vi) How close to 0 do we need to take x so that (x^2 + 6x + 9) < 9.0001 ? (vii) How close to 0 do we need to take x so that (x^2 + 6x + 9) > 8.9999 ? (viii) How close to −7 do we need to take x so that 1/(x+7)^4 > 10000 ? (ix) How close to −7 do we need to take x so that 1/(x+7)^4 > 100000 ?(x) How close to 0 do we need to take x so that ln x < −10000?(b) Use the definition of a limit to show that(i) limx→4 (x/2 − 2) = 0 (ii) limx→0 (2x + 9) = 9 (iii) limx→0 (x^2 + 6x + 9) = 9…
and let g(x) = Find the following functions and their respective domain x²-16 3x+2 5x-4 17. Let f (x) = %3D x2-16 Report domains in interval notation in the box provided. (a) (f- g)(x) (b) (fg)(x) To to To dinh chnu (e) () (2)
please send correct answer Q10
Determine the value of each of the following: D.) g(6)=
QUESTION 29 Kindly answer correctly. Please explain
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).
DISCRETE MATH PROBLEM
Can you please help me solve parts a, b, and c of the question attached as an image?
et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'? I would be very thankful if you could provide some explanation with the steps, thank you in advance.
4. Give the formula of a function that has each of the following domains. There are many possible answers. You can just pick one. (a) Domain x E (-0,00) (b) Domain x e (-0,3) U (3, 0) (c) Domain x e [5, 00)

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Question 1: Let e be a positive number, n e N, and r E R. In each of the following cases, find the value of o for which the given state- ment holds. Clearly show your workings. (a) a – 2| < d implies |r3 – 8| < E (b) |x – 1| < d implies < 1/n 3 1+ 2x