Recall that lim sup an = inf sup{am : m n}. n Assume that an 2 0. Show that lim an = 0 if and only if lim sup an = 0. n00

Q: 1. If f(r) = , do the following: %3D iii) What is lim

A: Given Function, f(x)=1x

Q: In(x – a) For a e R, which statements do hold for the one-sided limit lim ? 1¬a+ In(e² – eª)

A: Here we use L'hopital's rule

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

A: It is given that To prove that

Q: lim Va+l+v

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Q: Given that    lim x→4 (4x − 13) = 3,    illustrate Definition 2 by finding values of delta that…

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Q: In formally proving that limx→1 (x^2+x)=2, let ε>0 be arbitrary. Chooseδ =min(εm , 1). Determine the…

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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: can this be evalnated by using L'HOSpital's Ruie? lim Trxln(x) Itxen(x) (メ-)nx

A: The given limit can we evaluated by using algebra of limits.

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: Show: ( (|fl2 – €)² < lim sup E If (n) < (I|fl2 + e)² , Where ƒ E C(R | Z; , c) n=-N

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Q: Which of the following is an appropriate choice for o when proving the tollowing Ilm lim(z + 6z - 1)…

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Q: Show that the following: If r > 1, then lim Vr – 1 = 0.

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Q: lim w ln 2 In 1 3w

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Q: rasb then prove Lim Ja 9. that

A: If  0&lt;a&lt;b  then prove that limn→∞an+bnn=b

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: Show that if xn ≤ yn ≤ zn for all n ∈ N, and if lim xn = lim zn = l, then lim yn = l as well.

A: Given xn≤yn≤zn And also given lim xn=lim zn=l

Q: Evaluate the following limit. lim 5e ti-i+5tan tk 5t t+4+5tan 5t lim 5e li- -i+5tan'tk = t+4 (Type…

A: Solution:

Q: -1 Evaluate the limit: limx→o tan ( Inx) Select one: TT а. 2 b. –1 С. d. 00

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Q: Find the required limit or indicate that it does not exist. lim o+ (in (t6), 12 in t O 6i O Does not…

A: We have to find out

Q: In(7)/ In(x) (17 + 8x?) Find lim Limit is equal to (Type an integer.)

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Q: Use the definition of limits (ε-N) to show limn→∞ ((n/(2n−1)) = (1/2).

A: Find the subtraction value of sequence and limit

Q: Prove the following. (b) lim( Vn² +1 – n ") = 0 - (c) lim Vn? n) = } 1 2 ||

A: Prove the following limits

Q: 1. Use the definition to establish each of the followii (a) limx→2(2x– 7) = –3 (b) lim→1 T = (c)…

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Q: 5. Use L'Hôpital's rule to obtain the following limits: (a) lim r²Inr

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

Q: Suppose that lim,-00 f(x) = L, lim, y00 9(x) = M + 0, and f(n) = an, 9(n) = b, for all integers n…

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Q: a. Assuming that lim,(1/n') = 0 if c is any positive con- stant, show that In n lim n° п—00 if c is…

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Q: (a) Use MuPad to show that o for a> 0, for a= 0, lim = for a<0. Hint: Use assume to distinguish the…

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Q: Calculate: lim n =1 as follows: n = n% Innh Calculate lim In n and use this to determine lim Vn. = e

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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: 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: 3. Let fi R>R be a dipperentiabe quachon such and Fleo>7 if 2J0 the interual FCO) =d a) By using an…

A: Given that: F(x) is a differentiable function and it satisfies the following properties. F'x&gt;7…

Q: Calculate the following limit: lim(2r)ª sin(12 ).

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Q: Use L'H rule to find (a). lim zlng, (b). lim zn ez 1

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Q: If lim(x,) =5100, then (Xn) is bounded. Select one: O True O False

A: 1) Here, lim (xn)=5100 that is finite. That is the sequence xn is bounded. So, the statement is…

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: Suppose In(A5) In(tan z lim ( 1z))- 1Find the value M for cot 2z which A = eM

A: Here given as lnA5=limx→π2-lntanxcot 2x-15lnA=limx→π2-lntanxcot 2x-limx→π2-1

Q: Find the limit. (If the limit is infinite, enter 'co' or '-o', as appropriate. If the limit does not…

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Q: ven that lim f(z) = 1 and lim g(z) = -4, evaluate the following. lim (-4f(z) – 2g(x)) z-1

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

A: We have to solve given problem:

Q: Prove the following limit using the 8,e definition of lim,–→p f(x) = q: lim,→2(3x – 4) = 2

A: First we need to understand epsilon-delta definition to prove this limit-

Q: 7. Use L'H rule to find (a). lim zlnz? ez (b). lim Inz e

A: a.

Q: Find the limit of the following: lim In(3+ex)-In(9+x)

A: Use the properties of lograthim first than put the limit.

Q: Using the epsilon-delta criteria, our aim is to show that lim 3z – 2i = i. z→-i Which among the…

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Q: Evaluate lim S", r³dx . O 4 Does not exist.

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Q: Prove the following: If 0 < a < b, then lim,∞ (a" + b")" = b. %3D Civen L

A: Solution

Q: 6. Evaluate the following limits. arctan(z) – /2 (A) lim In (1-!)

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Q: Recall that to show lim,-→a f(z) does not exist, we need to find two distinct paths Yi and Y2 of…

A: |z|2=x2+y2So, xy|z|2=xyx2+y2Consider the equivalent equation limz→0xy|z|2=lim(x,y)→0xyx2+y2Along…