Could you do 43 and 45, thanks abouquence converges, find the limit.43. Assume 0 ≤ a ≤ b. Does the sequence {(a + b)¹/"}=1 diverge or converge? If the se-AQ44. Does the sequencek1{ (√²+)}k² + nk=1diverge or converge? If the sequence converges, find the limit.45. Show that if x is any real number, there is a sequence of rational numbers converging to x.*46. Show that if x is any real number, there is a sequence of irrational numbers converging to x.47. Suppose that {an)=1 converges to A and that B is an accumulation point of {an: nEJ).Prove that A = B.MISCELLANEOUS48. Suppose that {an)n-1 and (bn)n-1 are two sequences of positive real numbers. We say thatan is O(bn) (read as an is "big oh" of b) if there is an integer N and a real number M suchthat for n ≥ N, an ≤ M. bn. Prove that if {an/bn)n-1 converges to L # 0, then an is O(bn)and bn is O(an). What can you say if L = 0? Illustrate with examples.194PROJECT 1.1The purpose of this project is to prove that the sequence8{(₁ +9) Tn=1imply11TOY2.3To song T

"As you have posted multiple questions, we can provide the solution only to the first question as per our Q&A guidelines. Please repost the remaining questions separately."Question Number: 43Given Information:Choose such that .The sequence given isTo check:The sequence is converges or diverges.Find the limit if it is converges.Concept used:Cauchy second theorem on limits:For a positive sequence , if exists such that is finite and unique, then is also exist and it is equal to .Property: Consider the term of the sequence as .As is non-negative, is also non-negative, for any As is positive, is also positive, for any Therefore, is the positive sequence.Now, consider that .then, .Find the limit .As , it is the finite and unique as per the assumption of such that .By applying the Cauchy second theorem on limit, .i.e., the sequence convergent and the limit is .When and , the term of the sequence is .As is positive, is also positive, for any Therefore, is the positive sequence.Now, consider that .then, .Find the limit .As , it is the finite and unique as per the assumption of By applying the Cauchy second theorem on limit, .i.e., the sequence convergent and the limit is .If , then the sequence consists of only the 0 as its terms.It is the constant sequence and it converges to the same.i.e., zero sequence converges to zero which is taken for the place of .The sequence convergent and the limit is such that .