Question 1 Let be (0, 1). Prove rigorously that lim nb" = 0.

Q: 3. Let f(2) = if z + 2i, while f(2i) = 3+4i, is f(2) continuous at z = 2i? 2- 2i 4. Evaluate lim (az…

A: 3)Given:

Q: 2. (Part b, P2, Page 33; Give a direct e-N verification of the limit lim = 0. n o n+5 (Please refer…

A: To show  limn→∞1n+5 = 0

Q: 1) a)lim n(a (aa – 1) =?

A: Given: limn→∞na1n-1 We can write, limn→∞na1n-1=limn→∞ a1n-11n We get the indeterminate form

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

A:

Q: Question 5 Finding the limit: lim2 V12x + 3. O 9 O 5 O 3 O 7 O (E)

A:

Q: %3D 39. Prove that lim x*cos

A: To prove the given limit

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: :f and the functions fn are all monotone increas- If limn0. fn ing, must f be monotone increasing?…

A:

Q: Given that lim (4x - 1) = 3, illustrate Definition 2 by finding values of ổ that correspond to e =…

A:

Q: Given that lim (4x – 11) = 5, illustrate Definition 2 by finding values of d that correspond to e =…

A: limx→4(4x-11)=5we have to find the values of δ , when ε=0.5, ε=0.1 and ε=0.05

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

A:

Q: Question 7 of 10 ImaUna Evaluate 2√5x-2-√2x+4 lim

A:

Q: Question 5 Use the &-8 definition of limit to show that: ( a ) lim(3–2x)=-7 x→5 (b) lim x•cos-

A: Epsilon and delta definition.

Q: The numerical value of the lim 31-0 Question 1 n+3 nln (2) n+1 is equal to

A: Note:-Our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: 12. Suppose f € C[0, 1]. Prove that lim | f(a" )da = f(0). n>00

A:

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

A:

Q: Using the Sandwich theorem, what is the value of lim,[3 -2 cos(: n+1

A:

Q: Discuss the existence of the Limit Limit Cos 1/x-1 x-1

A: we have to check the limit of the given function. We will use definition of limit

Q: 2 for al 2 0. 39. Prove that lim x* cos %3D -

A: Given,

Q: Suppose that a, > 0 and lim a, = ∞. Prove that Sa, diverges.

A:

Q: The numerical value of the lim n+3 nln (2) "→ ∞ \ 11 + 1 is equal to

A: We need to evaluate the value of the following limit limn→∞n+3n+1nln2 Formula Used: limx→∞1+axx = ea

Q: Question 8. Prove that the limit: lim xVa 1.

A:

Q: Question 15: Suppose that the function f(r) is continuous at a ER and (rn) is a se- quence in dom(f)…

A:

Q: Question 12 Find the limit lim e2x+2h _ e 2x h-0 h O 2e* et

A:

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

A:

Q: Prove lim(Vn+1- n)=c 8. 0 =

A:

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

A:

Q: Ex 11.1.1 Compute lim r

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

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

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

Q: Given the sequence an 1 [" 1 dx. Then find lim an-

A: an=1n∫1n1xdx, then find limn→∞an

Q: Question 2: Assume limț¬ exists. Then lim al < o if and only if f(t) = 0(eªt) as t → o. Use this…

A:

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

A:

Q: EXAMPLE 4 Investigate the following limit. lim sin 5n x +0 x

A: In this question we have to investigate the given limit.

Q: Question 1 Prove the limit comparison theorem i. Suppose that an20 and b,20 and that an lim 7= Then…

A:

Q: Question 11 Evaluate the limit lim (ln(3x² + 2) - In(x² + 1)) O ln 3 O In 3 01 00 O ∞

A: To evaluate the following limit.limx→∞ln3x2+2-lnx2+1

Q: 3) see the attached file 3-the limit of this function cos x-(2 cos 2x -1) lim equal to 1-cos x A)2…

A: Given function is ; cosx - (2cos2x - 1) /(1-cosx) at x tendes to zero we find the limit .

Q: The result of the following sin 1² t limit is lim 1-0

A:

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

A: Given limit:

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: 8. lim In(+h)–1 A. 0 B. - C. 1 e D. e E. Does not exist

A: To solve the limit.

Q: Problem 3: Check by computation that if f(z) = , then lim ZEo+h)-{20) exists for all zo + +i. h→0

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Q: %3D In(1+x) 3.A) lim P-ove tha t マー0

A:

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: Problem 10.3.5. Suppose limn→∞ ¤n = c. Prove that lim→ xn = subsequence (ng) of (xn). ▼ Hint. First…

A:

Q: In(x), 5 s x s 16 f(x) A = lim n- o 2 i = 1

A:

Q: answer about the following questions brainly. limx-0 sinx(1-cox) 2x

A: To evaluate: limx→0sinx(1-cosx)2x2 Solution: limx→0sinx(1-cosx)2x2 On simplifying further we get:…

Q: cos 8-1 (e) lim 0→0 202

A:

Q: Express the limit lim > (5 sin(27x; ) + 8)Ax; over [1, 5] as an integral. n→ 00 i=1 Provide a, b and…

A: To find the solution

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: Question B.4. Find the limit of the following: n2-4n+4 1) lim n-2 tan2(n²+n-6) sin (z2-1) 2) lim…

A: Limit of the function

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.