Let an=n+1/n+3. Find the smallest number M such that: 1. |an−1|≤0.001| for n≥M 2.|an−1|≤0.00001 for n≥M 3. Now use the limit definition to prove that limn→∞ an=1. That is, find the smallest value of M (in terms of t) such that |an−1|<t for all n>M.

Q: For each natural number n, let the function fn: R → R be bounded. Suppose that the sequence…

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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: Evaluate the following limits lim as h approaches 0 (h-2)^2-4/2h lim as n approaches infinity…

A: The given limit,      limh→0h-22-42h Note:     Multiple questions are given, we are answer with only…

Q: Given that f,(x) = (n+ 1)(n+2)(1 – x)x" and that f(x) =0 for x E[0,1] 1. Show that the limit lim…

A: Note the result: If uniformly, is true. If point wise, can be true or false.

Q: 7. Let a, b, c, k E R. Compute the following limit kt e-s ds dt ct ax lim COs x -1

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Q: Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E…

A: Given Information -  f is a continuous function on R and is 2π periodic.

Q: Consider lim Vx + 3. x→1 (a) Show that if |x – 1| 2. (b) Find 8 > 0 such that if 0 < ]x – 1| < 8,…

A: The definition of limit: Suppose that f(x) is defined for all x in the open interval containing c,…

Q: Show that if Σ i=1 converges towards a € R , then lim ) a; = 0 i=N

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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: ) Let E be a non-empty compact (closed and bounded) subset of RP. Prove that every sequence X = (Tn)…

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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 (xn) be bounded, and lim (xn – xn+1) = 0. Let L = lim inf rn and H = lim sup xn. Show that every…

A: WE PROVE IT BY CONTRADICTION

Q: Given that lim n²(x– 5)" converges by the Ratio test. If x € Z then the numerical n- 00 value of x…

A: This is a problem of power series.

Q: In formally proving that lim (z² + x) = 42, let e > 0 be arbitrary. Choose ô nin). %3D m Determine…

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Q: For each n E N, consider xn n == X with x ER+. Define the sequence {gn} by 1 +2' 8n(x) = f(xn) (a)…

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Q: For each nN, consider x,, H (a) (b) 71 X with x E R. Define the sequence {gn} by 8n(x) = f(x) = 1…

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Q: Apply the ratio test to the series. n! 2n3 n=1 an+1 Find the limit lim an

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

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Q: Find an example of fn → f in measure, but for every x, fn(x) does not converge to f(x).

A: We have to find an example of fn→f in measure, but for every x, fn(x) does not converge to f(x).

Q: Compute the following limit. 1-e3 3x as x approaches 0

A: To find,  limx→01-e3x3x

Q: If a converges to a(z) then lim / a (2)dz= / a(z)dz n- 00 Y

A: True

Q: sin x 9. lim -0.1 -0.01 -0.001 0 0.001 0.01 0.1

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Q: What is the value of the limit lim,0o Vla,[ for the series Ey 00 (4n (4n O a. 4/3 O b. 00 O c. I O…

A: To evaluate the given limit.

Q: Prove by induction that 1 2. Prove that (xn) is contractive and thus convergent. Compute lim (xn).

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Q: Prove the following limit formula for e: e = lim (1+x)!/x

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

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Q: +9 for all n2 1. Prove that {r,} converges and find 2.1 4. (a) Let r = 4 and am+1 its limit.

A: Sometimes, a sequence is defined recursively. In that case, to test the convergence of the sequence,…

Q: Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.

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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 a function f(x) be continuous and positive for all x > 0. What does it mean to say that the…

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Q: 3. Find the largest n EN such that the limit h(-a2)- cos(x) lim exists, then calculate the limit.

A: Here, h(x)=∑n=0∞n!(2n)!xn And the series of cos x is: cos x=∑n=0∞(-1)n(2n)!x2n

Q: 3. Prove the following using the epsilon-delta definition of a limit. d. lim 1 = 0 X→-o X

A: We have to prove limx→-∞1x2=0 using the epsilon-delta definition of a limit.

Q: Prove the following using the definition of a limit. n - 1 (a) lim 1 n+on+ 1 (b) lim (2x² − 3) = −3…

A: Given a) limn→∞n-1n+1=1b)limx→0(2x2-3)=-3c) limx→2-1x2-4=+∞

Q: (e) lim Vn2 +2 -n = 0

A: Limit of the indeterminate from

Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

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

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Q: Let {xn} and {yn} be sequences such that lim yn = 0. Suppose that for all k ∈ ℕ an for all m ≥ k we…

A: Let {xn} and {yn} be sequences such that lim yn = 0. Suppose that for all k ∈ N and for all m ≥ k we…

Q: Let an for all n e N. Working directly from the Vn+1 definition show that limn- An = 1.

A: To evaluate limn→∞ nn+1: Here, an=nn+1 an can also be written as: an=1n+1n Simplifying this we get,…

Q: 5b) Let f (x, y(x)) = y(x) + 1 and y(0) = yo = 2. Use Picard-iteration and induction to show that YN…

A: Please go through the solution.

Q: Use series to evaluate the limits

A: Given: To find:  lim t→0  1- cos t - (t22)t4Formula:  cos t…

Q: (bn) is bounded, and b, #0 for all n E N. an lim bn = 0. Prove that lim (an) = 0.

A: Let an and bn be the sequence of real numbers with the property bn is bounded and bn≠0 for all n∈N.…

Q: Prove: If E an converges, then lim an = 0. n-00 Prove or disprove the converse of it.

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Q: Given that f,(x) = (n+ 1)(n+ 2)(1 – x)x" and that f(x)= 0 for × €[0,1] 1. Show that the limit lim…

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Q: If {an} is not bounded below, then lim tn where tn = inf{a;:k > n} 8.

A: Given a statement

Q: Given that lim n²(x- 5)" converges by the Ratio test. If x e Z then the numerical n- 00 value of x…

A: Introduction: The ratio test is also used to test the convergence of a sequence. According to the…

Q: Let an = n n + 1 . Find a number M such that: (a) |an − 1| ≤ 0.001 for n ≥ M. (b) |an − 1| ≤ 0.00001…

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Q: 65. Let S = E cn! where c is a constant. Σ n" n=1 (a) Prove that S converges absolutely if |e| e.…

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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.