If lim, +00 an 0 or lim,,00 a, does not exist, then an diverges. Select one: O True O False

Q: If lim an 0o and En1 a, diverges, then E b, diverges O True O False

A: taking the an and bn such that the given statement is False...

Q: (c) lim sup (an + bn) lim inf an + lim inf bn

A:

Q: Let a, be recursively defined by ao = 3, an+1 = a - an for n 2 0. Find lim an.

A:

Q: 5.7. Show that if lim f(z) = 0 and there exists a positive num- ber M such that |g(2)| < M for all z…

A:

Q: 8. Let BC2, C C2 and define if n is odd, В, C, An if n is even. What is lim inf An and lim sup A,?

A: For the solution follow the next step.

Q: 1 Find all values of a, such that ]lim a exits and finite. (x-1 х*-1 x->I

A: limx→11x-1-ax2-1

Q: Let a be any positive real number. Then * 1 lim In a – 1 True False

A:

Q: п+(-1)" (2n + 1) Sn

A:

Q: 00 Let a, = 1- /1 -. Show that lim an = 0 and that > an diverges. Hint: Show that a, > 2n n=1

A:

Q: Consider the following list of conjectures. Provide a shortproof for those that are true and a…

A:

Q: Suppose that lim, f(x) 0 and f(n) = an for all integers n> 1. %3D +00 Then, lim, 00 edn =0. Select…

A:

Q: Suppose fé Ra, b]. Prove that b- a lim b- f(x) dr. k=1

A:

Q: In(r² + 5) (a) lim In(r+2)

A:

Q: Present a “DELTA – EPSILON" proof, proving: prove th the de lim (3x – 7) = 5 X-4

A:

Q: Determine whether the following statement is true or false. If this statement is true, explain why.…

A: If limn→∞an=πthen an must be conversent. Reason- an is the fuction of n and when n is tending to…

Q: Which of the following is an appropriate choice for o when proving the tollowing Ilm lim(z + 6z - 1)…

A:

Q: In formally proving that lim (2² + x) (유, 1). = 42, let e > 0 be arbitrary. Choose 8 = min Determine…

A: Given limx→6x2+x=42 Consider x2+x−42=x2+7x−6x−42=xx+7−6x+7=x−6x+7≤x−6x+7 Hence, x2+x−42≤x−6x+7.....1…

Q: Show that if lim,→m aŋ = 0, then O(a, + 1) = 0(1). %3D

A: This is a problem of real analysis, complexity.

Q: Determine the limit of ? n/(2n^2)-1 and then write a proof of it using only the definition of limit,…

A: Let an=n2n2-1 Note that limn→∞1an=1limn→∞an. Therefore,…

Q: Let (xn) be a sequence of real numbers and let B be the subset of R defined by B := {:n e N}. Let…

A: Consider the provided question,

Q: define sequence a, by SELECT A A) B) lim a, =0 = 5-5t.5 C) -times %3D D) E) lim a F) lim a, = e

A: The given sequence is an, where an=555.....    (upto n times) We have to express this sequence as…

Q: 1. a) Do there exist sequences (an) and (bn) such that lim(anbn) = 1 and (an) diverges? b) If (an)…

A: See the attachment

Q: Suppose that a, > 0 and lim a, = ∞. Prove that Sa, diverges.

A:

Q: Prove the limit statements

A: Given that limx→31x2=13. Prove this as follows.

Q: #17: Evaluate the following limit. 7/2 dt lim

A:

Q: Determine whether the statement is true or false. If lim an = 0, then an is convergent. True False

A:

Q: 2-Jx-3 b) lim 7-х

A: b) Let I=limx→72-x-37-x

Q: Prove that lim n = p.

A:

Q: 1= 0. blem 2. Use the definition e-N to prove that the limit lim

A:

Q: Prove each statement using the s, 8 definition of a limit. a) lim(8x – 15) = 9 3+5x c) lim = 4 i 2…

A: Explanation of the answer is as follows

Q: Suppose that an > 0 and b, > 0 for all n N (N is a nonnegative integer). If we know that both an and…

A:

Q: Prove the following statement: if lim,∞ an = L and lim,∞ An = M , then L = M

A:

Q: Evaluate the limits of each of the following. 1. limn Vn + vn + 3 - Vn =

A: As Many Questions are given , I will solve only the first one We have to find : limn→∞n+n+3-n…

Q: If {a,} is decreasing and not bounded below, then lim an ylgn ihi

A:

Q: Exercise 2.6. Let An (-n-1,1+n-1) for n odd and An := (1 – n-1,2 +n=1) for n even. Calculate lim, An…

A:

Q: 30. Suppose that f, f', and f" are positive on R. Prove that lim f(x) = 00.

A: Let f, f' and f'' are positive on R. We need to prove that limx→∞fx=∞ Consider, the interval [a,∞).…

Q: If f : A → R, cER is cluster point and lim|f(x)| = L, then limf(x) = L, Select one: True OFalse

A:

Q: (b) From part (a), we know that lim an exists. Compute it. n00

A: Given that a1=3 and an+1=12+an for each n≥1 Also, the sequence an is monotonic increasing and…

Q: With regard to the limit lim x² = 25, Find a 8 such that |x² – 25| < 10¬3 if 0 < |x – 5| < 8.

A: We are given limx--&gt;5  X^2 = 25 We have to find delta , and we can find it by applying definition…

Q: ven that lim f(z) = 1 and lim g(z) = -4, evaluate the following. lim (-4f(z) – 2g(x)) z-1

A:

Q: Show that the following: Ifr> 1, then lim Vr -1 = 0.

A: We have to solve given problem:

Q: (c) lim (1+2x)/(2 In z)

A:

Q: 1. Evaluate the following limit. lim xIn 1

A:

Q: 1.9 Exercises 21 4. Suppose An = (-: m e N), ne N, where N are non-negative integers. What is lim…

A:

Q: Recall that lim sup an = inf sup{am : m n}. n Assume that an 2 0. Show that lim an = 0 if and only…

A:

Q: 7. Suppose lim nan = A +0 where an is not necessarily nonnegative. Prove >a,n diverges. n=1

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.