1. Define the following sets in extension: (i) {x:N|x< 4• 3 * x} (ii) {x,y:N|x < 6^ y< x•x-y} (iii) {x,y:N|x < 5 ^ y < 5•x +y} (iv) {x, k:N|x = k² ^ x < 10 • x}

Q: Find the indicated limit. a) V3x+5-V5 lim a) b) x*+2x3-15х2 lim х-з х3+2x2-9х-18 b)

A: As per our policy we have to do only single question.  A) 3/(2√5)  B) 12/5

Q: For all x, there exists y such that (y^2 <= x <(y+1)^2) when domain is N(natural number) I want to…

A: For all x, there exists a y, such that,  y2 ≤ x &lt; (y + 1)2   Given that x is a natural number,…

Q: 4) Define f: N→R as follows: f(1) = 3, f(2) = and for n > 3 f(n) = f(n- 1) + f(n- 2) %3D 2' Prove…

A: Consider the provided question, According to you we have to solve only question no. (4). (4) Given,…

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A: The mathematical induction is to be done in the following steps :  Let P(n) : be a statement .(1)…

Q: Show that the identity k n = n 72 n k k-1, olds for all integers 1 <k<n and use it to prove that n…

A:

Q: 5) lim (1 + 3)* X18

A: The given limit is: limx→∞1+2xx

Q: 10. lim( +1-V +1-)

A:

Q: (0 < * < 1) (-1<* < 0) (4) If f(x) = %3D expand f(x) in the form c,P,(x).

A: Give that: f(x)=-12, -1&lt;x&lt;012,0&lt;x&lt;1  Here L=(1+1)/2=1 Fourier  series expansion given by…

Q: 1) lim (1- – - 3,

A: The given limit is limx→∞1-3xx

Q: limit X----1    t+1/(t + 1)

A: Given function: t+1/(1+t)

Q: lim In x ( - 127/² + 1)

A:

Q: 20) Is the function given by f(x) ={x² - 4, for x <0 continuous at x = -2? Why or why not? for x 20…

A: solution:-f(x)=x2-4, for x&lt;0-2,     for x≥0check continuty at x=-2Left hand limitlimx→-2-…

Q: Find the length of the graph of f(x) = ² - In(x) for 1 < x < 10. 99 1 a) 4 -In(10) 99 + In(10) b) 4…

A:

Q: -7x2 + 7x + 2 14) lim x--- -16x2 - 7x +8 14) A) B) 1 D) -

A:

Q: Find * In(x² + 2x + 2) dx (x + 1)² In|1 + (x + 1)²| x +1 In|1 + (x + 1)2| x + 1 + 2tan-(x + 1) +c…

A: I am attaching image so that you understand each and every step.

Q: 2x2 – (3x + 1)Vx + 2 lim x-1 x - 1

A:

Q: d- lim x+1/(2x²+x+1)¹/² x →∞0 e-lim4|x+2/(x+2) X-→-2

A:

Q: Determine the recursive definition for the function f(n) = n + 2", where n ≥ 0.

A: The given function is f(n)=n+2n, where n≥0. We have to find the recursive definition for the given…

Q: x-2 j) lim -3x+2 3x+2x lim 0 x+4x k) x2 -3x+2 lim 2 x - 2x I) lim -x 1-x im m)

A: We will find the value of Given limit.

Q: 4x2+11х-3 (i) lim x→-3 9—х2 (ii) lim 2 -1 *-4 х36 6-х Vx2+9-V2x2+9 (iii) lim х-0 VЗx2+4-V2х2+4

A: We will solve the following

Q: 5. Using the limit method, show that every polynomial p(n) = a¿n² + ak- belongs to O(n*). ... + ao,…

A: Big theta notation is an order notation. It determines the growth rate of two growth functions. It…

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A:

Q: Find the limit of lim x(√64x²5 ∞0++* √64x²14) = = lim x(√64x²5 -5-√√64x²-14) ∞0++x

A: limx→∞x64x2-5-64x2-14

Q: x*+1 c. lim X+00 Vx+x-1 Vi+3x4 d). lim 2x+7 Vi+3x e). lim X-o 2x+7 f). lim (v4x² + 3x – 2x)

A: *As per bartleby guidelines we are allowed to answer only 3 sub-parts at a time. Kindly please…

Q: 7. lim (x +3) *+2| x--2- x+2 (a) -1 (b) (c) 1 (d) 8

A:

Q: c) lim (x – 7) x-7 f3x + 1, x< 5 lim x-5* (2x + 5, 5<x d)

A:

Q: 10) lim X+ 2x² +1 4X+2

A: Solution

Q: 8.) lim (-1+ x² – x³ + 3x*) 9.) x3-2x-1 lim x-3 co-X 2x2+3x+1 lim X+ 3x-5x+2 10.)

A: