Let A, = [n,n+ 1] and B, = [0,3+ [ for all n eN. 1. HL lim 42(A,) 2. PL B =2 MEN 3. PL An lim pz(A,) 4. PL B. = 3. NEN 1

Q: B. Prove the fillang tim Jn2 +n 4- (6) lim n-700 Lvntl trn

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Q: 42. Prove that if lim "O = 5, then lim h(t) = 15.

A: It is given that To prove that

Q: Suppose lim, An = 1. Prove that 1 lim E(-1)*a2k = 0. n-00 n k=1

A: Given that limn→∞an=1. To prove limn→∞1n∑k=1n-1ka2k=0.

Q: Show limn→∞(1 +1/2+1/3+···+1/n−lnn) exists

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Q: Q: Prove that Lim %3D n-a

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Q: 2. Evaluate the following: Va2-x² So Sa* x² ydydx а. xsin x b. lim-2cos x+e

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Q: 8. Let BC2, C C2 and define if n is odd, В, C, An if n is even. What is lim inf An and lim sup A,?

A: For the solution follow the next step.

Q: Prove that lim, Vn = 1.

A:

Q: Show that lim,--zo f (z)g(z) = 0 if limz¬zo f (z) = 0 and if there exist a positive number M such…

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Q: Suppose fé Ra, b]. Prove that b- a lim b- f(x) dr. k=1

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Q: 2.14 Show anx" + ... a12 + ao lim n an bn .. provided that b, # 0.

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Q: 2+ 1 ** Evaluate lim - X+ oL dt A B none of these 1- e? 1 4

A: Given that,  Now taking the integral term,

Q: 8. Calculate the following limits. In z lim → 2²-1 (b) [ 1³-1³ 0 e + cos2-1-2 lim

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Q: (2) Show that Jo for a > 0 but not for a = 0. lim 818 √ne-na² da a lim √ne-na2 748 da

A: We have to show that,   limn→∞∫α∞ne-nx2dx = ∫α∞limn→∞ne-nx2dx  for α>0  but not for α=0.

Q: In formally proving that lim (2² + x) (유, 1). = 42, let e > 0 be arbitrary. Choose 8 = min Determine…

A: Given limx→6x2+x=42 Consider x2+x−42=x2+7x−6x−42=xx+7−6x+7=x−6x+7≤x−6x+7 Hence, x2+x−42≤x−6x+7.....1…

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: in Q5. Find the value of the limit: lim in — е 2 2+1 Z→e 2

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

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Q: Compute the limit below. 3(In a)²+3(lnz)+2 lim (In a)²+1

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Q: 1) @) Prove that lim an= o if lal <1 and a ER 6) show fon any X with Ix) <1, lim nxn=0- that

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Q: 15.) Show that lim((2n)/"): 1/m =1.

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Q: Show that 2n lim n+2 = 2,

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Q: 2. Let (X, A, μ) be a measure space. If {E} CA and E₁ E₂ E3 ..., then prove **** that (UE)= lim…

A: A limit is a value that a function approaches as the input approaches some value. Limits are…

Q: Prove that lim (1/ nP) =0, p> 0.

A: Given the equation which to be prove

Q: 4x In z (a) Evaluate lim エ→1( (1- 2)-- -1) C.

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 19 let ɛ > 0 be arbitrary. Determine & as a function of e 2 In formally proving that lim -x + 8 1/4

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Q: min 1) In formally proving that lim (x + x) 42, let e > 0 be arbitrary. Choose & %3D Determine m. m…

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Q: 5.6. Prove that: (b). lim z0 Z (а). lim does not exist, = 0. - ||

A: The given problem is to prove and evaluate the following complex variable limits.

Q: 1. Use an e-proof to justify each of the following. (-1)"=0. (a) limn→∞ n (b) limn→∞ = 0. n (c) lim2…

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Q: In(x)-In(a) 14. lim %3D Xa X-a (A) = (B) In(x)NT 92 anevni art

A: Given, limx→ alnx-lnax-a

Q: #17: Evaluate the following limit. 7/2 dt lim

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Q: Prove that lim [V4n² + n – 2n] = -

A: Use rationalize to solve

Q: e) Finally, prove that lim az-2 does not exist.

A: Use left hand limit and right hand limit to proof it

Q: 19 , let e > 0 be arbitrary. Determine & as a function of e In formally proving that lim + 8 2

A:

Q: 10. Prove that lim x2 = 4 using the epsilon-delta definition of a limit. %3D X-2

A: limx→2x2=4

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: 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: 1. Use L'Hospital's Rule to compute In(1+x) – lim | x2 2 Compute

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Q: Suppose that an > 0 and b, > 0 for all n N (N is a nonnegative integer). If we know that both an and…

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Q: Prove that lim r = 2 using the e-8-definition. Hint: VI - 2 = VT + 2

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Q: e -e lim is z0 In(1+ 2x)

A: Given limit:

Q: show that If an > 0, n = 1, 2,. .... In+1 In+1 lim inf 14x <lim inf n ≤ lim sup Van ≤ lim sup nx In…

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Q: 1 Use the definition of the limit (ɛ – 8) to show that 1 lim = i. z-i z

A: We will prove the given limit.

Q: Show that the following: Ifr> 1, then lim Vr -1 = 0.

A: We have to solve given problem:

Q: please help thank you

A: We need to prove the given statements, which are,

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: 1.9 Exercises 21 4. Suppose An = (-: m e N), ne N, where N are non-negative integers. What is lim…

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Q: 3 In(z) Carefully define lim 21 2-I = 3, including phrases with e and d. %3D

A: Note that lnx&lt;=x-1  I have used that inequality to prove the statement

Q: 1. By introducing suitable limiting processes in 1 8(x) = 27 eikr dk, show that (a) 8(x) = 1 1 lim…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.