Statement S: For any positive integer n, lim =h%3D2とラ2 (a) (Basis step) Prove bye-d definition that statement S is true for n.= 1, that is lim r' = 21.%3D2시+ 2시z- 2.(b) Prove, with reasons, if k is a positive integer that r*+- 2k+1| < |x||x* - 24|+2*|x – 2|.-(c) Use (b) and prove if Je- 2 <1 then2시 + 2시2 - 21.|r*+ – 2*+1| < 3a*

Answered: Image /qna-images/answer/98487673-58ca-445f-879c-2cdb00987cdd.jpg

Answered: (a) (Basis step) Prove by € – d…

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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mathematically before reading the proof. (It might be helpful to read the discussion that follows Theorem 2.4 or to look at Appendix A.) THEOREM 3.2 THEIS a function that is defined on I except possibly at the point c. Let I be an open interval that contains the point c and suppose a) The function f has limit L at c if and only if for each sequence {Xn) in I\ {c} that converges to c, the sequence (f (x,)} converges to L. b) Suppose that there are sequences (x,} and (y,} in I\ {c} that converge to c such that {ƒ(xn)} converges to L, and (f(yn)} converges to L2. If Li + L2, then the functionf does not have a limit at c. Proof Let P be the statement "the function f has limit L at c" and let Q be the statement "for each sequence {xn} in I\ {c} that converges to c, the sequence (f(xn)} converges to L". A proof that P Q will be left as an exercise. To prove that Q = P, we will prove the contrapositive. Suppose lim f (x) # L. By the negation of the definition, there exists e>0 such that for…
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18

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(a) (Basis step) Prove by € – d definition that statement S is true for n = 1, that is lim r 2!. (b) Prove, with reasons, if k is a positive integer that |r+- 2k+1| < |x||x* - 24|+ 2 |r - 2|. (c) Use (b) and prove if a – 2| <1 then |r*+1 - 2*+1| < 3- 2 + 2*|r – 2|.

(a) (Basis step) Prove by € – d definition that statement S is true for n = 1, that is lim r 2!. (b) Prove, with reasons, if k is a positive integer that |r+- 2k+1| < |x||x* - 24|+ 2 |r - 2|. (c) Use (b) and prove if a – 2| <1 then |r+1 - 2+1| < 3- 2 + 2*|r – 2|.