Q/prove that if bn, an are real and positive and 0 < lim b = bn <∞0, n ∞ a= lim sup an then ab= lim sup (an bn). Does this remain true if the 11-0 11 00 requirement of positivity is dropped ?

Q: show that the lim sup(an+bn) (as n -> infinity) infinity)

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

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Q: (b) Show that if lim sn = +o∞ and lim tm > -x, then lim($n + tn) = +0.

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Q: 14 Find the limite by using L'Hopital Rule: X-I ①Lim 3x²-x²-2 sin 5x X 3 ③Lim 1-2 Sinx

A: Step 1: L'Hôpital's Rule: States that if the limit is in the indeterminate form of like…

Q: Evaluate the limit n 1 2/3 lim 2 V3 3n n→∞ 1+ 3 + (k – 1)2v3' - k=1 3n

A: Given : limn→∞∑k=1n11+33+k-1233n2 233n Therefore, limn→∞∑k=1n11+33+k-1233n2…

Q: xCosx-In(1+x)] х-Тапх, limx-ol limx-o[ a) х-Sinx c)

A: a ) given limit is as below:

Q: Suppose there exists No such that sn No. (a) Prove that if lim sn +00, then lim tn (b) Prove that…

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Q: 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,)…

A: Here non of the options are correct.

Q: Show limx→∞ xne−x = 0 for all n = 0, 1, 2, . . . . To get started notice that when x ≥ 0, all the…

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Q: Suppose that lim an = 5 and lim bn = 9. n-0 n→∞ Determine the limits. lim (an + bn) = n→∞o lim a =…

A: We are given that limn→∞an=5 and limn→∞bn=9.We need to find the given limits.

Q: Let fn f, where if 0<x<! if <a<i -x if 0 <x<1 fn (x) n-2 (2 - x) 2 and f(x) %3D if x = 0 Then lim…

A: 2

Q: 1. Let f(z) = In z. )inthe defiiio (b) Find f(1) by the formula (In1 (c) Draw the conclusion that…

A: To derive the properties of the exponential function using the relation : (ln(x))'=1/x

Q: 1. lim z-sinfinity (az+b) ³/(z+d)³ 2. lim z->1 (1-z bar)/(1-z) does not exist

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Q: Show that if lim,→m aŋ = 0, then O(a, + 1) = 0(1). %3D

A: This is a problem of real analysis, complexity.

Q: 2 Prove that limx* cos== 4 = 0. %3D - x→0

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Q: lim (w,x,y,z)→(0,0,0,0) w²x+2yz² w² + x² + y² + z²

A: Given: lim(w, x, y, z)→(0, 0, 0, 0) w2x+2yz2w2+x2+y2+z2

Q: Let the region be bounded by the curves y= 3 /x and y = 9, and x= 36. i) ( )Draw this bounded region…

A: We have to find the volume of region.

Q: डणि2X Lim 2. x²

A: We have to find the limit of given function.

Q: anx" + + a1x + ao An 4. If f(x) = bn with an #0 and b, #0, show that lim f(x)% = ... %3D .. + b1x +…

A: As per the guidelines I am supposed to do the first question among multiple questions. So I am doing…

Q: 4. ·| cotl t• dt 0. lim limitinin değeri kaçtır? n In(x + 1) a) 3 b) 1 c) O 2 d) 4 Boş bırak

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Q: Prove that if limn→∞ (an) = 0 and (bn) is bounded, then lim n→∞ (anbn) = 0. One cannot use limit…

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Q: (5) Prove: (b) lim 230 c) lim 140 (Im²) ². 12₁²2² does not exist (Rez)² +Z1 0

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Q: 37. lim (In 2x – In(x + 1)) X00

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Q: (a) (Basis step) Prove by € – d definition that statement S is true for n = 1, that is lim r 2!. (b)…

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Q: 3. Let f(x) = x and P = {0, nn = 1}. ... + f("1) and that lim Lf(P) + k = k(k +1)/2) 1 Show that…

A: Given the function f(x) = x &amp; a partition of the interval 0,1 as P =0,1n,2n,3n,...nn=1. First we…

Q: Show that an is decreasing and bounded from below where an+1 = (1+a^2n)/3 and a1 = 1. Then determine…

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Q: x² +3x+2 15) limx-2 xt-16 %3D 32 1 c) 3 d) DNE e)non a) b) 1

A: Solving the problem by finding the limit of the problem.   NOTE :: As per our Expert guidelines we…

Q: Exercise 7. Evaluate lim vn(Vn+1- Vn). n00

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Q: Find: L . -im Z÷1+1 L Z²+1 2 Z²_1 im Z10 +1 7/0 14-Z 6 Z+1

A: (a). The given limit is: limz→1+iz2+1z2-1

Q: 6. Calculate lim . Use L'Hopital's Rule and show all steps. x→∞ e

A: To find the limit of the function using L'hospital rule

Q: 5. If limn→∞ an 1 (a) limn→∞ bn (b) limn→∞ A and limno bn = B 0, prove B' an = bn B.

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