1. Define the following sets in extension: (i) {x:N|x< 4• 3 * x} (ii) {x,y:N|x < 6^ y< x•x-y} (iii) {x,y:N|x < 5 ^ y < 5•x +y} (iv) {x, k:N|x = k² ^ x < 10 • x}

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: 2. Let X, Y be nonempty sets of real numbers that are bounded above. Let X + Y = {x + y:x € X and y…

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Q: (b). Compute:  (i) Limit points. (ii) Interior of the set (iii) Closure of the set (iv) Boundary…

A: In the given question we have to calculate the Limit points , interior point ,closure , boundary…

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Q: 18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34…

A: Solve the following problem

Q: Find the limit given by 1 1 + n + 2 1 lim n→0 (n + 1 2n O In 3 O In 2 1 infinite + +

A: SOLUTION-

Q: EX: Show that the telation ? >' is PartiaL oldering set of Por R-

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Q: 17. Use the limit process to fnd the anea of the negion bounded by the graph of f(x)=2x43 ón the…

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Q: verify whether the limit to infinity of n(n + 1)(n + 2)−¹(n + 3)–¹ is equal to one.

A: By finding the limit, we prove the limiting value of the function is equal to 1

Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

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A: Given,

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Q: 9. Use the squeeze theorem to evaluate lim(x – 1)²ecos (Hint: X-1 -1< cos s s 1, x # 1) ズー1

A: If h(x) &lt;=f(x) &lt;=g(x)

Q: 2 Find the ares bounded by the function f(x)--r+2, the s-axis, 0 and 2 in cach of the folloreing w *…

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Q: Show that has a continuous extension to x = 2, and find that extension.

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Q: Compute f'(x) using the limit definition. f(x) = 1 – 26x-!

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Q: 1. Compute the following limits : In (2x+1)-2X x2 Inx a) lim b) lim X-1* (lax)

A: Solution:To find the limit of the followingFormula:ddx(ln x)=1xddx(xn)=nxn-1ln ab=b ln a

Q: 8. Using squeeze theorem, show that the following limit (x – 1)² In x = 0. lim (x,9)→(1,0) (x – 1)²…

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Q: If x > 8 and x n+1 - - for nEN. Then the limit of x is equal to 11 X 21

A: Given that, If x1&gt;8 and xn+1=2-1xn for n∈ℕ. The objective is to find the limit of xn.

Q: limit as h---0 for the funtion ((e^2h)-1)/h using L'Hopitals

A: We need to evaluate the given limit using L'hopitals rule. As per L'hopitals rule, if we get 0/0 or…

Q: 4) State the limit definition of the derivahve. f'(x) =

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Q: 4. Show that if A is closed in Y and Y is closed in X, then A is closed in X. Verify it and make…

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Q: 8. )Prove that lim Vr = 2 using the e-8-definition. Hint: VI – 2 = x- 4 VI + 2 Puouo thot thoro…

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Q: Express the limit as a definite integral. 5 lim 2 cAxK, where P is a partition of [2,11] ||P|→0k=1

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Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

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Q: a. Inscribe a regular n-sided polygon inside a circle of radius 1 and compute the area of one of the…

A: Consider a n-sided polygon inside a circle of radius 1. A polygon is the union of the isosceles…

Q: Evaluate the limit using L'Hopital's Rule. 3 3 lim e -1 00 3 3 2

A: Please refer the attached image for complete solution.

Q: Let m and n be two positive integers greater than 1. ecos (α"). am m then the value of (1) is n If…

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Q: If the limit of f(x) as x approaches cis 0, then there must exist a number k such that f(k) < 0.001.…

A: Explanation: If limx→cf(x)=0, the for any positive ∈, (Like 0.001) there is a positive δ such that…

Q: 2. Explain why there does not exist a measure Space (X,S, M) with the pro porty %3D

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Q: Compute the limit n+0 (1+)2n o0(1+)²

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Q: 5 Set up an expression for f Inx dx as a limit of sums. 3

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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

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A: To choose correct option from given question.

Q: lim (1+e *)*

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Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

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

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Q: 1. Prove the following limits x 4y4 G) lim (x,y) (0,0) (x²+y4)3 = Does not exist

A: Solution is given below:

Q: X be a limit point of set ECR, then every o infinitely many points of terval containing

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Q: Prove that for any bounded sequences (xn) and yn we have lim sup(xn + Yn) < lim sup(xn) + lim…

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

Q: not receive points. 1. Compute the following limits:

A: Given: (a) limx→1xn-1x-1(b)limx→∞x1/x(c)limx→∞x2+1-x+1

Q: Compute fallcwing limits if they Dlim 8+ iz exist Z ナ4 2) lim Rez. こ20

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Q: Evaluate the limits in Exs. 46-51. Vz-2 46. Lim Ans. . 47 Lim Ans. 0. (1 – 2*)! (1-) (z -4z + 5' -…

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Q: 2. Let {fn} be a sequence of nonnegative measurable functions and fn (x) → f (x) on a set A of…

A: Given, {fn} is a sequence of non-negative measurable functions and fn(x)→f(x) on a set A of finite…

Q: 1.Determine the line limit {an} = 37 (2cos ! en

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Q: 2. Compute each limit below using limit laws. 7+3x (a) lim 7+3x (b) lim r-4 4-x 7+3x 4-X

A: we have to find the following limit

Q: 5. Conclude on the limit of f(x, y)=y as (x, y) (0,0) along paths y = mx and

A: In this question , we can find the limit along the given path, Firstly we can find limit along the…

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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