Let fn(x) = n²x²(1 – x) on [0, 1]. Does lim,oo fn(x) exist for every r E [0, 1]?

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

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

Q: Prove that

A:

Q: Discuss limit(n →∞)fn on B and C for  fn(x)=x/n; B=E1; C = [a, b] ⊂ E1 fn(x)=(cos x + nx)/n; B=E1

A: Note: Since you have not mention anything about the set E1, We are considering it any sunset of real…

Q: Find the global maximum and minimum of the continuous function f(x) = x²-8 ln x on [1,4]. %3D

A: Given:F(x)=x2-8lnx        1,4

Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

A:

Q: x2 + 0x – 4 lim x-2 x - 2 Enter I for oo, -l for -co, and DNE if the limit does not exist.

A:

Q: Consider the function f(x) = vx for xE R and 0 <x < 32. Prove that lim,-s f(x) = 4. %3D

A: Given: f(x)=x for x∈ℝ and 0&lt;x&lt;32 we have to prove that limx→16f(x)=4

Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

A:

Q: > If f is a continuous, decreasing function over [1,0) and the lim f (x)=0, X00 then :)dx is…

A:

Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

A:

Q: Express the limit as a definite integral. n 7 lim 8 1 + i = 1 n- o n dx

A:

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: d x"J,(x)=-x"Jns(x) d, n+1 X.

A:

Q: Let            fn(x) = nx/1 + nx2  (a) Find the pointwise limit of (fn) for all x ∈ (0,∞).

A: Given : fn(x)=nx1+nx2; To find the point wise limit of this function for given values of x: We have…

Q: Find the limit. 3 - ti + In(t)j k t-3

A:

Q: Let fn(x) n²x²(1 – x) on [0, 1]. Let e > 0. Does lim,0 m ({x € [0, 1]|fn(x) > e}) exist?

A: Given that fn(x)=n2x2(1-x) on the interval 0,1 The objective is to show whether…

Q: 1. Find the limit of the following functions: x2/3+x-1 lim e x2/3 + cos² x 1- cosx lim- -+0 sin2x

A:

Q: (-cx² + 2x, (sin x + cos x, x> n x<π 9) Prove that lim (x* sin+5) = 5 10) f(x) = What value must c…

A:

Q: Use residue theorem to evaluate the integral ∫T/(1+x2)2dx lower limit of the integral is -∞ and…

A: Given: ∫-∞∞71+x22dx (1+x2)2=0 1+x2=0⇒x=±i only x=i will inside the region. (-∞ to ∞ is the upper…

Q: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

A:

Q: Suppose f(x,y) = sin-' (6 – 2x2 – y²) and g(x,y) = 4x2 – y2. Determine the continuity of the…

A:

Q: 1. Evaluate the limits. e' cos (t – 4) dt - 4 (а) lim x2 x2 – 4 e2 + 2 (b) lim x00 e2x + 100

A: Solve the following

Q: Let fc Cx)= Vx if osxs 4 c.(x-4)+2 if x>4 denotes a real constant. - where C Show that lim fe (x) =…

A: The function fcx=x,              if 0≤x≤4cx-4+2   if x&gt;4. Where c is a real constant. We have to…

Q: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: a(sin 2a+cos a sin 2a) lim 1-cos² a a→0

A:

Q: 5 The function f (x, y) =| x² cos² is homogenous in x and y of degree,

A:

Q: )Use the Sandwich Theorem to show that lim cos x-1 =0. x2 Find f'(x) for f(x) = sec(vx2 + 1).

A:

Q: cos x In(x – a) | 40. lim In(e* – e) X→a+ -

A: Use L hospital's rule when of the form type of inderminate 00.

Q: Find the linearization L(x) of the function at a. f(x) = cos(x), a = 3n/2 %3D

A: We have to find the linearization L(x) of the given function.

Q: - Evaluate the limit a-Lim(x/(x-1) – 1/ Inx) b- lim [In(cos(x)) c-2-VX-+4 /sin'x x>1 x>0 d- lim x+1…

A: Limit values

Q: Find lim r(t) when 5 In t r(t) = (: 2 tan¬lt, e-2 t 2 tan-t, e 1. limit = (2т, —2, 0)

A: Given    r(t) = 2tan-1t , e-2t , 5 lnttwe need to  find  the limt→∞ r(t)=?

Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

A: The given function is, fx=-x2+4x-3 The orthonormal system is, O; i→,j→ To evaluate limx→-∞fx  and…

Q: Find the limit or prove that it does not exist lim (2,y)→(0,0) T° cos x + 2y sin y x2 + 4y2

A: We will evaluate the limit limx, y→0, 0x3cos x+2y3sin yx2+4y2  using polar coordinates.

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

A:

Q: 4. If R(T) is the autocorrelation function of a stationary process {X(t)} with no periodic…

A:

Q: Let fn → f, where if 0 <x < L3 (3 – x) if 1 < x<1 { fn (x) п-3 if 0 < x < 1 - and f(x) if x = 0 Then…

A:

Q: 5 The function f(x, y) = | x' cos is homogenous inx and y of degree

A: Given that: f(x, y)=x2cos2yx-y2+x3y

Q: 'e in + = "T 2n Find lim a

A:

Q: 2t – 2 t2- 1, Let r(t) = - î - tan (t) ĵ+ In(2t) k. %3D (a) Determine the domain of r. (b) Compute…

A:

Q: Find the Wronskian for y, = sin 3x , and y2 = cos 3x.

A: The Wronskian, W, of two functions, f and g is calculated as follows. W=fgf'g' , where, f' , g'are…

Q: 3. Evaluate the limit r(t +h) - r(t) lim h→0 h for r(t) = (t-1, sin(t), 4).

A:

Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

A:

Q: F(x) Let F(r) = | 3e" dt. Find lim if it exists. If the limit does not exist, explain why.

A:

Q: 1. Show that for all f e L'(R) lim da = 0.

A:

Q: Find lim e dt 0. 1 1/e

A: We find out limit of given function. See 2nd step.

Q: Prove that lim n 00 a" f (x) dx = 0, for any f E R[0, 1].

A:

Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

A:

Q: 2. Find a function f(x) such that n n 1 n+ i i=1 n n i=1 3. By evaluating an integral, compute the…

A: