) If f(x) = (0 <* < 1) (-1 < * < 0) expand f(x) in the form c,P(x).

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

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

Q: 2) Evaluate the limit at x=1 for: x* + 7x2 – x – 2 f(x) x² – 5x + 4

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Q: If lim 2 4xk (1-3xk')Axk, a = - 3.b=3· Then the correct expression for the limit as integral is: k…

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Q: 1) Find the dexivative by the limit process. fex> 2X - 3 3X +4

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Q: 2. Find the limits. limx→-1 sin(x² + 3x + 2) x³ + 1 M

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Q: Find the Taylor polynomial of degree 4 for cos(x)cos⁡(x), for xx near 0: P4(x)=P4(x)=  Approximate…

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Q: The limit as x approaches infinty of (x+3^x)/(x^2-3^x) Evaulate using L'Hospital's Rule providing a…

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Q: lim b-> -0.1x e 1.

A: Let I=limb→∞∫1be-0.1xdx

Q: (a) Show that the limit x*ycosy lim 12 (xy)→(0.0) 2x* +y² does not exist.

A: Student has specifically asked for the solution to question (a).

Q: evauluate the limit lim x→0 (sin2 (3x))/ x2

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Q: Show that >n"x" diverges for all x + 0. n=0

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Q: 2. Show that X H→ In x is increasing on (0,0) and that In 1 = 0:

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Q: Ql(i) Prove that Lim t* ... /2n²+1 2n²+n]

A: The series is Lim12n2+1+12n2+2+......+12n2+n=12

Q: Limit n---infinity for the function nln((1-(3/5n))/4 using L'Hopitals

A: Evaluate the given function  as follows.

Q: If f is a continuous function on [0, ∞0) and | f(x)dx converges, then I f(x)dx diverges. 3 Select…

A: Given that, f is a continuous function on [0,∞). And, ∫3∞f(x)dx  converges. Then,…

Q: Compute limit as x goes to infinity of (cos(x))/(x)

A: We find: limx→∞cosxx We solve this limit by using Squeeze theorem. Squeeze theorem states that if…

Q: Show that Hint: Use to obtain In[|x|] √√2/T. 1 - cos(sx) x² 8 1 - cos" x x² lim n-√√n 00 2 π dx |s|…

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Q: Discuss the existence of the Limit Limit Cos 1/x-1 x-1

A: we have to check the limit of the given function. We will use definition of limit

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: Why does the limit as x goes to 0 of cos(1/x) not exist? Can use two different sequences:…

A: Solution

Q: What is the limit as x goes to infinity of (ln(x3 + 9) - ln(2x3 + 17x))?

A: Use quotient rule of logarithm to reduce the function as shown,

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

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Q: Let fn(x) = n²x²(1 – x) on [0, 1]. Does lim,oo fn(x) exist for every r E [0, 1]?

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Q: Determine the infinite limit. limit as x approaches 3 from the right side of ln(x2-9)

A: We have ln(x2 - 9 ) if we take x as √10 which is approximately equal to 3.16, then, x2 - 9 =1…

Q: Find the limit. sinx cos 4x lim x-0 x + x cos 5x A) 3 B) 0 C) D) Does not exist 4/5 -/2

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

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: Let fn: R --> R be defined by : fn(x)= x/(1+nx2), For all n >= 1. a) Show that {fn} converges…

A: step:-1 Theorem:- Let fn be a sequence of functions on I such that for each n∈ℕ, f'n(x) exists for…

Q: 2. The limit of the following function equals: x2/3 + x-1 lim x--00 x2/3 + cos2 x

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Q: 1) Show that the limit doesn't exists : ysinx 3) lim (x.y)-(0,0) x²+y2

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Q: 9 -x2 if x 1 .What value of c makes g(x) continuous at x = 1? Show all conclusions. 2 Vx-4 Using…

A: For a function to be continuos, the left hand limit should be equal to the right hand limit.…

Q: Find the limit of tan^-1(e^(x^2)+x^2+1/x) as x approaches negative infinity

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Q: Compute the limit as x approaches zero of f(x)=(x^2+3x)/(x^2-x-12)

A: Given : Function f(x)=(x2+3x)(x2-x-12)   To find : Limit of given function when x approaches to…

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 f is bounded and integrable in the interval [a, b], (x) cos nx dx3D0. show that lim

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Q: Prove that lim x cos 4 = 0. X-0 -1 s cos(4/x)s 1 = -x s cos(4/x) s x*. since lim (-x*) = x-0 and lim…

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Q: - Find the limits using L-Hopital's Rule: cos 3x 1 (а). im $ x→0 X* (b). lim (e - 1)*

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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: 18. Compute, from first principles only, without using L'Hopital's rule eh – 1 lim h-0 = 1

A: given is  limh→0  eh-1h=1we know    eh  = (1+h1!+h22!+h33!+.............)limh→0  eh-1h = limh→0…

Q: x-3x+5 by X-0 4x2+2x+3 10. Find the limit: lim means of the L'Hopital Rule

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Q: 4. Letf(x)=x22/5 - 19x². Determine the largest n for which fis n-times continuously differentiable.

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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: (b) Use the sandwich theorem to show that limy0x°cos - = 0. 1 In order to apply the sandwich…

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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: Q1:Find the limit using L Hopital rule In (csc x) lim xT/2 (x – (1/2))²

A: Given limx→π2ln(cscx)(x-π/2)2 We have to find limit by using L Hospital rule L'Hospital's Rule…

Q: 3. Compute the limit algebraically Vx + 3x2 lim 6x 1

A: Solution: The objective is to find the limit of the given function

Q: Express the limit as an integral. Š (5(x; )² – 3(x; )³)ax over [0, 2] lim

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Q: Define the sequence of functions {fn(x)} on [0, 1] by { 2n2x; 2n(1 — пх); т. fn(x) : 0; n Find f(x)…

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Q: 2. If (xn) and (Yn) are sequences of real numbers such that lim (xn) + 0 and lim (xn Yn) exists,…

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Q: In (x + h) – In x (c) lim h→0

A: To find the limits  Here variable is h , so assume x as constant. 0/0 form , so use L-Hospital's…