Let a, be a dounded nonnegative sequence prove that: 1 lim sup n-0 (1+xn) 1 lim sup(1 +n) 1 lim inf (1 + an) n00 n 00

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

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Q: for all 2 Use the definitton of a Umit of a sequence {an 3ne r to show that lim (on-3 { an 3nen to…

A: We will prove the given statement.

Q: 3. Prove that lim sup (a, + b) ≤ lim sup a,, + lim sup b, and lim inf (an + b) ≥ lim inf a,, + lim…

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Q: inf{a,bn :n e N} > (inf{an : n e N}) × (inf{b, : n € N}). (1) Can the inequality in (1) be strict?…

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Q: Which of the following subsets of R satisfies the following property: If (x,) is a Cauchy sequence…

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Q: lim x → 0

A: →When limit is of the form 00 or ∞∞ then we use L'Hospitals Rule  →In L'Hospitals Rule , take…

Q: Let (an) and (bn) be two bounded sequences of positive real numbers. Prove that lim sup(anbn) (lim…

A: Given two bounded sequences of positive real numbers  and .We have to prove that, .To do this, use…

Q: 3x + 2x + x+ 1 Let Ax) = What types of asymptotes does Ax) have? Explain. .2 x* +x-2

A: To find the asymptotes of f(x).

Q: kle consider the sequence (Un) defined for 17, 2. = Un-1 + 3 Up-2 with loro and real numbers…

A: Here we use the definition of monotonically increasing sequence in step (2) and in step (4) we use…

Q: Find lim (1+ I-X 4 T

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Q: Compute lim 1+-

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Q: Evaluate 1 1+x)s - (1-х) x)8 lim X - |00

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Q: Let C be a fixed positive constant. If {bn} is a sequence of nonnegative numbers such that lim (as…

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Q: (d) (i) Define: sequence of real numbers {x} converges to the limit x = R. 3n+1 7n-4 (ii) Use the…

A: d.1. Definition:          A sequence of real numbers xn is said to be converges to the limit x∈ℝ…

Q: 2 ( - 12/² + 1) In X 11) lim

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Q: 10* – 5* – 2" +1 Evaluate lim-

A: The L'Hospital rule can be used to evaluate the limit of a function if the limit is obtained in the…

Q: The first term of a converging sequence {an} is given by, a1 = 0 and the remaining terms can be…

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Q: Consider the sequence an = 3" + (-4)". We can a) lim 8+40 an = -∞ b) the sequence is increasing c)…

A: To analyze the sequence an​=3n+(−4)n , we need to understand the behavior of each term as n…

Q: Show that if lim supn→∞an = L for a finite real number L and > 0, then an> L+ ε for only finitely…

A: The objective is to prove the statement: If limsup an=Ln→∞ for a finite real number L and ε>0,…

Q: 3. Consider the sequence {2+2} 7 lim n-con² + 4 +2=_ Definition of limit: (Substitute the given…

A: The limit is 2

Q: -1 Suppose that (an), has an >0 for all n. Show that lim sup a, = (liminfa,)¯. Suppose (an)=1 and…

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Q: Suppose {Pi} and {qi} are sequences of real numbers such that lim pn = p and lim qn = q. Give an e-N…

A: suppose pi and qi are sequences of real numbers such that   limn→∞pn=p  and limn→∞qn=qSo by…

Q: Let an = Let an = 5n100 - 2nn n! + ln(n) 23n+3 +6n 7.8 +15n2 Then lim an = n→∞ Then lim an = n→∞

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Q: 2. Use the "E-N" definition of a limit of a sequence to prove that 2n + 3 1-4n lim 818 4 = -- 2

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Q: Apply the theorem: /Let (XN) be a sequence of positive real Numbers such that Li 2:= lim (XN+1/XN)…

A: Given sequences: (a): (b): (c): (d): where  and  satisfy We have to find the limit of the given…

Q: Find the limit given by 1 1 + 1 lim n + n + 2 2n O In 3 O In 2 O infinite

A: First we will find given limit by using riemann sum

Q: Evaluate 1 n + 22 lim n (n + 1) n+ 2. n + 2n -1

A: Introduction :We have to calculate the limit of given function :

Q: If the sequence {a} is strictly decreasing then lim a does not exist 8444 6) lim 2 = -00 →+8 a lim…

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Q: Find the lim x -9+ 6 + x DNE -00

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Q: n! a) Find the limit, to infinity, of the sequence xn (2n+1)!

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