Let (X,) be a sequence of real numbers. If for some x e R, (x,) converges to both 7x + 2 and 2x - 13 then O a. none of the listed statements is true O b. (X,) converges to 19 O c. (x,) cannot converge to both 7x + 2 and 2x - 13 since the limit of a sequence, if exists, is unique O d. (x,) converges to -19

Q: 1. Define when a sequence X = (x,) in R is said to converge to x E R. Then prove that the sequence X…

A: The given sequence is: xn=3n+12n+5

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A: Proof

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

A:

Q: (d) Let x* = 1/2(2*), and what is the limit of x* as k 0? What is the order of converge for x?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 2. a. Prove using the formal definition of a limit that the following sequence converges: -n3 + 2n –…

A: This is a problem of sequence.

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N&gt;0, there exists a δ&gt;0,…

Q: 3п Use the ratio test to determine whether converges or diverges. (2n)! n=4

A: Apply series ratio test and find whether the series is converges or not.

Q: For each natural number n and each number x in (-1,1), define f_n(x)=sqrt((x^2)+(1/n)) and define…

A: Uniformly continuity :- Let I⊂ℝ and for each n∈ℕ, let   fn : I →ℝ be  a function then the sequence…

Q: For each nN, consider x,, H (a) (b) 71 X with x E R. Define the sequence {gn} by 8n(x) = f(x) = 1…

A:

Q: Show that the sequence f.(x)} converges to a limit function (that you should determine) for each XEI…

A:

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: Given that lim n2(x-5)" converges by the Ratio test. If xEZ then the numerical 1- 00 value of x is…

A:

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn– 1+ Yn – 1) and y,= Vx – 1 Yn –1 V n 2 2.…

A:

Q: Suppose that f is a bounded non-negative measurable function. Show that there exists a decreasing…

A:

Q: Find the limit of the sequence an={sin(n)/n}..if it exists.

A: Take limit n tends to infinity for finding the limit.

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n²x; if x E (0, ) = (x)"f 0; otherwise Find the…

A: we have sequence of function fnx=n-n2x ; if x∈0,1n0;           otherwise pointwise limit of sequence…

Q: Suppose that (a, } is a monotone sequence such that a, S11 for all n. Must the sequence converge? If…

A:

Q: } be a sequence of real numbers. } is unbounded, then {xn} has no limit. 2} is not monotone, then…

A: We have given few statement . a) If xn is unbounded, then xn has no limit. b) If xn is not monotone,…

Q: Let {x„}" and {ym}*=1 be convergent sequences with limits x and y, respectively. Suppose that all…

A:

Q: Suppose x1 : 1/2 and xn+1 := xn2. Show that {xn} converges and find lim xn.

A:

Q: Give an example of a sequence of continuous functions: fn: [0, 1] R that is pointwise convergent but…

A: Consider the function fx=0, x is irrational in 0,1mn, x is rational in 0,1. This function is called…

Q: Show directly from the definition that if (x„) and (yn) are Cauchy sequences, then (X, + yn) and…

A:

Q: Given that lim n2(x- 5)" converges by the Ratio test. If x EZ then the numerical n- 00 value of x is…

A:

Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

A:

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A:

Q: 3. Show that if x1 = 1 and if In+1 Van + 2 for n E N, then the sequence (xn) is monotone and…

A:

Q: Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: Let a 1 and define an+1 = V4an -I for n2 1. Show that the sequence {an} converges and find the…

A:

Q: Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential…

A:

Q: 9n + 14 Determine whether the sequence an converges or diverges. If it converges, find the limit. In…

A: an=9n+147n+9 if  limn→∞an=∞ then the series is diverges otherwise, it is converges

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

Q: In = Vn (Vn +1- vn

A:

Q: 28. yn = ne/n

A:

Q: Let f : A → R and (xn),(Yn) are two sequences converges to c such that Xn, Yn # c, Vn E N. If…

A: We understand this statment by the help of an counterexample .

Q: Given two sequences (fn)neN, (9n) nEN C C[0, 1] of continuous functions on the closed unit interval…

A: Given two sequences fn, gn contained in the interval 0,1 of continuous function on 0,1 by…

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn- 1+ Yn – 1) and y, = Vx, – 1 Ya-1 V n 2 2.…

A:

Q: Given a sequence ( xn ), prove that (xn ) converges to zero if and onlyif the sequence ( |xn| )…

A: From the given information, it is needed to prove that:

Q: For any n N, define the function Xn RR by : if x ≥n Xn(2)= 0, if a <n. Show that the sequence of…

A: We will use definition of limit of sequence of function.

Q: Suppose that {a,} is a monotone sequence such that a, 8 for all n. Must the sequence converge? If…

A: suppose that an is a monotone sequence such that an≤8 ∀n it is known that a sequence is said to be…

Q: Let (x,) be a sequence of real numbers. If for some x e R, (x,) converges to both 7x + 2 and 2x - 13…

A: Which of the following is correct?

Q: Suppose that (an) and (bn) are convergent sequences of real numbers with limits an → a and bn → b,…

A:

Q: (a) Suppose that f : X → Y is a continuous function and {xn} is a Cauchy sequence in X. Does {…

A:

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: Let (X,) be a sequence of real numbers. If for some x e R, (X,) converges to poth 7x + 2 and 2x - 13…

A: which of the following statement is true.

Q: Given that lim n²(x- 5)" converges by the Ratio test. If x e Z then the numerical n- 00 value of x…

A: Introduction: The ratio test is also used to test the convergence of a sequence. According to the…

Q: 40. d, = Vn + 3 – yn

A:

Q: Prove that a sequence of positive real numbers {r„}1 tends to 0 if and only if the sequence {},…

A: We have to prove a sequence of positive real numbers xnn=1∞ tends to 0 ⇔ 1xnn=1∞ tends to ∞.

Q: Prove that the limit of the sequence V2, y2v2, V 21 V2, \/ 2\/2/: /2/2, exists, and find it

A: Given the sequence               2,22,222,2222,.....

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.