Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

Q: Let x1 > 1 and  xn+1 = 2 − 1 / xn for n ≥ 2.  Show that ( xn ) converges and find its limit.

A: Given - x1>1 and xn+1=2-1xn for n≥2 To prove - xn is convergent and also find its limit .

Q: Q2) Evaluate the limit at x = -2 for: x 1 2x2 4x - 8 S(x) x² - x- 6

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Q: Let f e C(0,1). Suppose that lim f(x)= lim f(x) =+∞. Show that f acheives its minimum in (0, 1).…

A: Given: f∈C0,1 limx→0+fx=limx→1-fx=+∞

Q: Evaluate the limit V9n1 + 13n' = 5n – 10 lim =V25n16 +5n2 -9n Also, find the asymptotic value of…

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Q: Suppose x1 : 1/2 and xn+1 := x(n/2). Show that {xn} converges and find lim xn.

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Q: (d) Let x* = 1/2(2*), and what is the limit of x* as k 0? What is the order of converge for x?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Express the function n, lim n→+∞ 4k² in closed for and then find the limit. n³

A: We will first write this function in closed form then we will calculate the limit of the function.

Q: Let f(x) = 2x² + 2x + 3. According to the definition, f'(x) = lim h→0 (Enter the limit in reduced…

A: the soluton is following.

Q: {VI2n + 8 – V12n} converges or diverges. If it converges, find the limit.

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Q: Sequences {xn} and {yn} such that they are both convergent and {xn+yn} maps to 5

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Q: If fn is a uniformly equicontinuous sequence of functions on a compact interval and fn → f…

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Q: Use Picard's method to find y1, y2, Y3. Determine the limit of the sequence {yn(x)} oo for

A: Introduction: A differential equation is an equation that contains the derivative of an unknown…

Q: Let {fn(x)} : „x ER and > 0. 1+ nx Show that ift>0, the sequence {fn(x)} converges uniformly on

A: Given: To explain the given statement as follows, To show that if t>0, the sequence fnx converges…

Q: Suppose that a, > 0 and converges. lim na, = 0. Prove that Ea, n 00

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Q: Evaluate the limit /9nt + 13п3 — 5n - 10 lim n-10 -V25n16 + 5n² – 9n Also, find the asymptotic value…

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Q: 3^n+2 / 5^n converge or diverge?

A: f(n)=3n+25n=3n325n=9(35)n The sequence converges as 35<1.

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: Prove that the sequence {f(1/n)} has a limit if |f'(x)| < 1 for all r € (0, 1]. I

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Q: If the function f is continuous on [0,00) and | f(x)dx is convergent, then there is a possibility…

A: If the function fIs continuous on [0, ∞) and ∫1∞f(x)dx is covergent, then there  is a possiblility…

Q: Suppose x1 : 1/2 and xn+1 := xn2. Show that {xn} converges and find lim xn.

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Q: Determine if the given is convergent or divergent. (x²+1)² n Zn=1 (n²+1)²

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Q: Suppose that F(u) denotes the DFT of the sequence of f(x)={1, 2, 3, 4}? What is the value of F(14)?…

A: Given: F(u) denotes the Discrete Fourier Transform of the sequence f(x) = {1, 2, 3, 4} To find F(14)

Q: If lim,f(x) = 0, then f(x)dx converges Select one: O True False

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Q: Give an example of a sequence of continuous functions: fn: [0, 1] R that is pointwise convergent but…

A: Consider the function fx=0, x is irrational in 0,1mn, x is rational in 0,1. This function is called…

Q: find if diverges or converges. if converges, then find its limit cos²(8n) An 8n-1

A: Given - an = cos28n8n - 1 To find - find if diverges or converges. If converges, then find its…

Q: 1. Consider the sequence Xn  = √n + 1 − √n, n ≥ 1. Prove that (xn)n is convergent. Find its limit.

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Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

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Q: Let an 2n+4n-3 Using the definition of the limit, prove that lim,∞ an 3n2+2n+1* 2/3.

A: Given that, the rational function  an = 2n2+4n-33n2+2n+1 To prove that limit of function is limn→∞…

Q: If lim=c>0, then both convergent and both divergent. ylgn O Ihi O

A: If limn→∞anbn=c&gt;0, then ∑bn,  ∑an both convergent and both divergent. (a) True (b) false

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

Q: Give an example of a limit of the form 0/0 that convergesto 1 as x -&gt; 0.

A: Given: Rx=fxgx is a function such that limx→0Rx is of the form 00 and converges to zero. To…

Q: show that the limit of a^n/n! is 0 as n approaches infinity

A: Gien To find limn→∞ ann!

Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n2x; if x E (0, ) fn(x) : otherwise SHOW THAT:…

A: The given sequence is fnx=n-n2xif x∈0,1n0otherwise First to find the limit function of this sequence…

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn- 1+ Yn – 1) and y, = Vx, – 1 Ya-1 V n 2 2.…

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Q: 1-e * xp- By using the limit comparison test, the integral 1. x2 + 1

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Q: Evaluate the limit V9n4 + 13n³ – 5n – 10 lim -V25n16 + 5n² – 9n Also, find the asymptotic value of…

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Q: Construct a sequence of continuous functions fn : R →→ R such that 0 < fn S 1, limn-o Jo fndx = 0…

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Q: 2n+1 4. Use the definition to prove that ) converges.

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Q: Suppose f and g are continuous functions. Provide an example such that | g(x)dr diverges 0. adf(x)dr…

A: Follow next step

Q: b) Use the Comparison Theorem with f(x) = - to show whether , g(x)dx is convergent or divergent.

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Q: 4. Let (#1, y1) = (0,0) and let (n+1, Yn+1} = (2+ yn, 3 – Tn) for n > 1. Prove that (Cn, Yn) is a…

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Q: Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn (x) = ,x +1 +…

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Q: Give an example of a sequence of continuous functions: fn : [0, 1] → R that has no pointwise…

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Q: 1-e X x2 +1 X By using the limit comparison test, the integral

A: Using, limit comparison test

Q: 7. Determine if f In (x") dx is convergent or divergent. Justify your results.

A: Note:- We are entitles to solve the first question, unless specified. To determine:- ∫0eln(x2) dx is…

Q: ar converges if and only if lim a, = 0. r->0 ア=1 True False

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Q: find if diverges or converges. if converges, then find its limit cos² (8n) an where n 2 0 8n-1

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Q: ) Evaluate -The limit (lnx) as approaches positive infinity (a 1 Zero e O Infinily

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.