5. Compute the maximum, minimum, in fimum and supremum (whenever these exist) for the following sets, (i) {1+ 1+(-1)" :n e N} (ii) {x : x € Q and x² < 2} 6. Use the c- definition of limits of a sequence to cstablish the following limits (i) lim,+∞(,) = 0 (ii) lim,,+∞12) = 3 n+00 n+1 (iii) If a > 0 then lim,, yo0(na) = 0

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: 1) Find the Limit lim/→0 3x²h-7th+h h

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Q: 5. Prove that the set of points at which sequence ef measurable real-valued functions converges (to…

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Q: 3n²+ 2n 32 (a) n² - 11 (b) (7) Hint: 4" = (2¹)²

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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: Calculate the following limits of functions using the sequential character- ization from Theorem…

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Q: Find the limit

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Q: (4+h)2-16 lim h→0 Compute h

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Q: Use the definition of the limit of a sequence to prove that (-1)"n' lim = 0. n2 + 1

A: To prove limn→0(-1)nnn2+1=0

Q: 1) 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: 6. Show that the following sequences of sets, {C& }, are non-increasing, (1.2.17), then find lim-»∞…

A: * SOLUTION :-  (6) Given sequence of sets , {CK}, non- increasing, (1.2.17),

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

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Q: with lim an = 0 and lim f(x) = L,then lim f(an) = L. x0+ Show directly from definition that if an is…

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Q: 3. Let (Sn) be a sequence of nonnegative real numbers, and suppose lim s, = 0. Prove that lim S, =…

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Q: Find the limit: (1 -4)" lim n- 00 in

A: Here we have to find;  y=limn→∞1-2n3n

Q: Evaluate each of the following limits: (a) lim (2+³) (b) lim (2t-³) t→∞ ∞047 (c) lim (2+³) t→0+

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Q: (e) limVx ) lim -+(-1)") X-00 n→0 n n²-12345 x2 +9 (g) lim n +1 (h) lim x∞x3 +1

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Q: 4 – x lim x - 4 5- Vx² + 9

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Q: 6. a) State why the limit of f(x) = √5x - 1 does not exist for x

A: Please refer to the image below

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

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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: Establish the x² - x + 1 lim +2 3 %3D By using the 4. x+2 € - 8 definition of limit or the…

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Q: a. For each of the following find the limit if it exist. I. II. x- 1 lim X-1 x2 +x- 2 lim (2-* + 3)…

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Q: 3. Prove that for any bounded sequences (n) and yn we have lim sup(n + yn) ≤ lim sup(x) + lim…

A: We have to prove that limsupxn+yn≤limsupxn+limsupyn. Let xn and yn are sequence of R. Let limsupxn=x…

Q: compute the limit using (a) table of values and (b) substitution

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Q: 3. Which of the following sequences are bounded: (i) () (ii) (n+ (iii) (1-) (iv) +1)

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Q: 15. a) Find the limit. x²-9 lim x--3x²+2x-3 b)lim √25+h-5 h

A: (a) let H=limx→-3x2-9x2+2x-3

Q: (2n+1)!+(2n+2)! )! (2n+3 lim

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Q: Compute the following limits: 2 3 3h h-2h-2 (a) lim (d) lim h-0 √h+3-√3 (b) lim x-0 12-4 -2 12+10…

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Q: 4. Assuming the limit exists, explain what ∞0 means and explain why an ∞0 limit could be anything…

A: To discuss indeterminates and limits.

Q: /49+h - 7 lim h→0 h

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Q: x² + kx-24 x-3 lim- 6. Given that 3 exists, determine the value of k, and then evaluate the limit.

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