If f is continuous and {an} is a sequence such that the limit lim an = L n➔ oo exists, then lim f (an)= f (L)

Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

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Q: Let x, = 3. Using the algebra of limits and standard results on limits to establish/justify if the…

A: Consider a sequence an. Then if the limit limn→∞an exist then the sequence converges otherwise the…

Q: (Зп + 1)! Find the limit of the sequence an = If the (Зп — 1)! limit does not exist, type "Diverges"…

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Q: Let bn = an+1. Use the limit definition to prove that if {an} converges, then {bn} also converges…

A: Given that, bn = an+1 By Using definition of Limit,  To prove that if an converges, then bn also…

Q: Does the sequence {a,} converge or diverge? Find the limit if the sequence is convergent. an = 2…

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Q: lim nsin(2/n) 2→+∞

A: Given query is to find the limit of the sequence.

Q: Prove that the sequence given by 2.4. 6.(2n) Un = 3.5.7.…(2n + 1) converges and find its limit.

A: Given query is to prove that sequence converges.

Q: Prove directly from the definitions that if g : ℝ → [1, ∞) is a function so that limx → c g(x) = L,…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Find the limit of next. sequence using the theorem for Limits 2n+1

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Q: . Suppose real sequences (S,) and (t,) are bounded. (That is, that their ranges are bounded sets.)…

A: Given the real sequences sn and tn are bounded. We have to show that The sequence given by sn + tn…

Q: an=n?(cos( Find the limit of the sequence (a,), if it is convergent. Otherwise, please type…

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Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+o n

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Q: If a sequence c1, C2, C3, ... has limit K then the sequence eº1, e^2, eC3 fact together with…

A: We are given the sequence,                        .

Q: 22: Establish the divergence of the sequence x, =(-1)"

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Q: Prove limit of n to inifinity of 5/n = 0; where An = 5/n.

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Q: Assume (fn) and (gn) are uniformly convergent sequences of functions. (b) Give an example to show…

A: Given : Assume (fn) and (gn) are uniformly convergent sequence of functions.

Q: Let {an}o be a sequence with positive terms. (a) Prove that if lim van= ∞, then La, diverges. (b)…

A: According to Bartleby guidelines, I can give you a solution for only the first three subparts. so, I…

Q: Let {bn} be a sequence whose limit is 3. Show that if B is an upper bound of this sequence, then 3 <…

A: We will use the definition of limit.

Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

A: Let f is continuous at c and xn be a sequence such that limnxn=c Since, f is continuous at c, for…

Q: Let f : (0. 1) → R be bounded but such that lim f does not exist. Show that there are two 1-0…

A: Let xn be a sequence in 0,1 such that lim xn=0  As f is bounded so, fxn≤M for some M&gt;0 So, by…

Q: Theorem 1 states that if lim x→∞ f (x) = L, then the sequence an = f (n) converges and lim n→∞ an =…

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Q: Exercise 8. Let {an}-1 be a sequence such that an lim L+0. n-00 n Prove that {an}1 is not bounded.

A: Given that &lt;an&gt;is a sequence such that  limn→∞ann=L≠0 To prove: &lt;an&gt; is not bounded.

Q: Find x so that x, x + 2, and x + 3 are consecutive terms of ageometric sequence.

A: We have to find the value of x such that x, x+2, and x+3 are consecutive terms of a geometric…

Q: 3) Prove that lim(xn) = 0 if and only if lim |x| = 0. Give an example to show that the convergence…

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Q: Determine whether the sequence converges or diverges. If it converges, find the limit. If it…

A: We have to find

Q: ii) Prove that if a sequence XnXn converges to a limit l, then any subsequence of XnXnalso converges…

A: Just use definition of convergence.

Q: what is the limit of this sequence limn->Infinity (n*sin(π/n))?

A: The objective is to find limn→∞nsinπn

Q: Use series to evaluate the limit. -1x2 x - In(1 x) 2. lim x0 +3

A: Known fact:

Q: Consider the following sequence: en + en (a) Show that the sequence is monotone and bounded (and…

A: Find the limit of the sequence

Q: an = cos

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Q: Use series to evaluate the limits

A: Given: limx→0ex-e-xx we know that…

Q: has limit e Use this fact if a sequence c,2, C3, ... has limit K then the sequence e, e2, e,...…

A: To find the limit of a given sequence, bn.

Q: Prove that sequence {an}= {(-1)n} does not have a limit.

A: According to the given information it is required to show that {an}= {(-1)n} has no limit.

Q: Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].…

A: According our guideline i solve only first.

Q: 1. Prove the sequence {a,} = (1 +-)" Converges to the limit e*, what about the sequence {b,} = (1…

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Q: Prove using the definition of a limit that the sequence

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Q: 4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0 %3D

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Q: Does the sequence {an} converge or diverge? Find the limit if the sequence is convergent. Зд 2 = sin…

A: topic - sequence and series

Q: prove that if X =CXn) is bounded increasing sequence of real ConVergent and Lim (xu) - Sup [An ineN]…

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Q: If {xn} is a bounded sequence on R. Prove: lim n→+∞

A: Given  sequence xn is bounded  we want  to show that  limn→∞xnn=0

Q: Show using sequences of functions that 2 lim e dx = 0.

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Q: 2. Determine the convergence of the sequence given by: an+1 = 1+ 2 an ao = 9 Find the limit if it…

A: 2. To Determine: The convergence of the sequence given by: an+1=1+2an, a0=9.   To Find: The limit if…

Q: If lim,→00 an 0 then an converges. Select one: O True False

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Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A: The given sequence is,

Q: Let f and g be continuous functions from R to R. Define the set C {x €R: f(x)=g(x)}. If (x,) is a…

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Q: is converges, then lim a, Lan =. If n=1

A: Given that ∑n=1∞an converges then the limn→∞an=∞ If the series converges then the sequence of the…

Q: Use series to evaluate the limit. X-In(1 +x) lim

A: Find your answer below

Q: If a sequence c1, C2, C3, . . . has limit K then the sequence e1, e2, e3, . .. has limit e“. Use…

A: Answer

Q: 2. Let r,LERand f:(-0,r) → R be a function. Prove that lim f(x) =L if and only if, for every X--…

A: Let r,l∈ℝ and f: -∞,r→ℝ is a function. If limx→-∞f(x)=L then for every sequence xn in -∞,r such that…

Q: Find out if the sequence is convergent or divergent. If convergent, find limit. lim x→∞ cos^2x   /…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.