6n + 2 3. n→∞ 2n + 1 3. Using the definition (i.e. N, E), prove that lim

Q: (b) Let a1 = 1 and an+1 = lim, an- /am+2. Prove that a, is convergent and find %3D

A: The recursive relation represent a bounded above increasing sequence

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

A:

Q: Find the limit infimum and limit supremum of the following x₂ = Yn where Yn is bounded. Хр n lim…

A:

Q: Using the Nth term test for divergence 5 1 n=1_2n +14 5 diverges since lim is equal to 1 2n+14 O 1/3…

A: We can answer the question as below.

Q: 2) Use the e-8 method to prove lim (3x - 1) = 2 Given e > 0, there exists a d such that if 0 < |z-a…

A:

Q: 3. Show that the largest value of o that can be used for some e > 0 if lim (2+ 5x) = 17 is エ→3 anx"…

A:

Q: lim In (z\x2 + y²) P - (5, 5, 1)

A:

Q: Evaluate I (n +) lim as n → 0 Vn[ (n + 1)

A: We have to evaluate limn+32n⋅n+1 as n→∞ that is limn→∞n+32n⋅n+1 We know that n=n−1! Hence,…

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

A:

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: Determine whether the following statement is true or false. If this statement is true, explain why.…

A: If limn→∞an=πthen an must be conversent. Reason- an is the fuction of n and when n is tending to…

Q: 1. Prove lim, n+2n³+7 2 0, using the definition.

A: I have used only definition of limit of sequence

Q: find the Value of the limil [z(そ-e 241 lim

A:

Q: 14k 14 as a definite integral. 8 Express the limit lim Σ n→00k =1 9 + + 121

A: Known fact: Let f be a continuous function on a,b. Then ∫abfx dx=limn→∞ ∑i=1n fxi ∆x where ∆x=b-an…

Q: to evaluate the limit. 3 1 v 78. lim x→ x Jo im In(1 + e') di | In(1 + e') dt 2

A: Answer is  1/2 Explanation is given below

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

A: True

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: 4-IS. 2-x s Mx) s 2Cos(x) for all x then find: lim f(x)

A: It is an application of sandwich principle of limit

Q: 1. (f Use "e- N"-statement to prove that lim n 1 [Hint: Ve > 0,3N(e) = > 0, Vn > N, we have n- %3D

A: We have to show that given limit of function converge to 1-

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

A:

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+n Prove using the definition of the limit that limn→∞ 1-2n 2

A:

Q: If(z)- 니 < e. By definition Show that: lim, »1+i = - а. b. 4 lim = 2i. %3D 2- 1-i z2

A:

Q: (7) ifa lim has a point that the equalio- prove PCz,y) is above L and only if 3x +2y >4

A:

Q: 18. Find (if exists) * lim (x,y)→(0,0) (4 + (x² + y² ) In(x² + y² )) O 1 O Dose not exist - O 4

A: Limit of the two variables function : Let f be a function of two variables .The limit of f(x,y)…

Q: Prove that lim vn+1= 1. n-00

A:

Q: 10.2.29 Because 0.5 1. e 3+(1/3"-1)

A:

Q: Find the lim inf (xn) and lim sup (xn) of Xn = √1 + n² 2n - 5

A:

Q: 1 Assume that lim xn +oo. Prove that lim n-00 n +o0. n-00 k=1

A: Given limn→∞xn=+∞

Q: Extend the result proved in Example 2.5.3 to the case |b| &lt; 1;that is, show lim(bn) = 0 if and…

A: We have to show lim(bn) = 0

Q: 8 Evaluate the limit lim [(z-1)(3- 2i)+1-31] and use the definition of the limit (e-8) z+2+i to…

A:

Q: (iii) 2 an must converge if lim an 0. n =0 bn (iv) 2 bn must diverge if b, > 0 and lim = 4. n->0…

A: The solution are next step is

Q: 7. Compute the limit to infinity: 4-2+3r+1 a lim 7+3r-r+ 12

A:

Q: Examplesa using the definitin of limait to .2 2. Prove i lim Z =Z

A: In mathematics, a limit is the value that a function (or sequence) approaches as the input (or…

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…

A:

Q: Find the lim inf (xn) and lim sup (x₂) of Xn = COS nπ

A: I have calculated the limits

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 Q5. Use the precise definition (i.e., via the e 8 notation), prove lim х-2- (х-2)3 = -0.

A:

Q: 2. Prove that 3 lim 0. (HINT: This proof needs a ɛ-M type argument.

A: To prove that limx→∞3x=0

Q: 4x 7 lim 1+ Compute the limit: x→0

A: here we put value of limit then we calculate

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: 3. Prove the following using the epsilon-delta definition of a lir a. lim (2x – 1) = 3 x→2

A: We will prove the limit using definition.

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

A: Using, limit comparison test

Q: Exercise 3 Prove that t2 lim (1+ (19)

A: The given expression is in the form of one to the power infinity. Let L=limp→∞1+t2pp+12. In the…

Q: Using e – N description to prove that lim = 2. n2 +1

A:

Q: Q1(i) Prove that 1 1 1 Lim en? +1 V2n²+2 2n²+n] II

A: I AM DOING PART 1 ONLY BEACUSE THERE IS NO SPECIFICATION OF "n" given in the question , neither…

Q: 7. Suppose lim nan = A +0 where an is not necessarily nonnegative. Prove >a,n diverges. n=1

A: