For each description below, find examples of sequences {an}neN and {b}nen such thatan→0 and bn ∞ and such that(a) anbn →0(b) anbn → ∞(c) lim→ anbn does not exist, but anb is bounded.(d) Given c ER, anbn → c.

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Answered: For each description below, find…

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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Consider the sequence b₁ = 1+tan n . In order to apply the squeeze theorem and determine the limit of b, as n→ ∞o, we need to find two sequences, a,, and C,,, such that an ≤ b ≤ Cn and lim an = L = lim c,, for some real number L. Which n 11-00 11-00 of the following pairs of sequences will satisfy these conditions? Select the correct answer below: O an = 1-2 and C₂ = 1 + 2 1-π an = ¹ and C₂ = 1+x O an =-1-2 and C₂ = −1 + 2/ O an. = -1+ and C₁ = ¹+ 2 Content attribution W S * X H pand # 3 80 E D C $ 4 888 R FL % 5 LC. V T G command ; + 11 { [ لا opti
B. Show directly from definitions that if sn and to are sequences so that lim-00 Sn = -8 and limn - tn = ∞o, then lim-00 Sn + tn = 00.
3a. Prove from the definition of limit that lim bn +L also.
Consider the function f : R → R, and define the sequence {an} such that an = f(n). True or False? If lim n→∞ an = L, then lim x→∞ f(x) = L If true prove it, if false,provide a counterexample (i.e. a specific function for which the statement does not hold).
2. Let {an} be a bounded sequence in R and ƒ : R → R be a continuous and increasing function. Prove that lim sup f (an) = flim sup an n→∞ n→∞
Consider the function f : R → R, and define the sequence {an} such that an = f(n). True or False? If limn→∞ an = l, then limx→∞ f(x) = l. If true prove it, if false, provide a counterexample (i.e. a specific function for which the statement does not hold).

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For each description below, find examples of sequences {an}neN and {b}nen such that an→0 and bn ∞ and such that (a) anbn →0 (b) anbn → ∞ (c) lim→ anbn does not exist, but anb is bounded. (d) Given c ER, anbn → c.