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

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: Suppose f is differentiable on the set of real numbers R and a E R. Let F(x)= f(xª) and G(x) =…

A: F(x)=f(xα)F'(x)=f'(xα)ddx(xα) , [chain…

Q: Find Lim as x approaches 0 of the function cos(4x-1)/2x

A: Given: limx→0cos4x-12x To find:  The limit limx→0cos4x-12x

Q: 3. Find the domain of the following functions: a.) f(x) = V1-x Inx b.) f(x) = Vx - 1lr

A: solutionsince the given (3)(c)f(x)=1-xlnx(b) f(x)=x-l ln(x)use have to find the domain of the…

Q: tan x 1. Differentiate the function: Y = ea* In x

A:

Q: Let f(r) = In r Find and simplify f"(x).

A: Please refer the attached image for complete solution. THANK YOU.

Q: If y =f (x) is a function in x then prove that f(a + nh) ="C,f(a) +"C, Aƒ(a) +"C,ƒ(@) + +"C, A"…

A: Given: y=fx is a function in x Prove that: f(a+nh)=C0nfa+C1n∆fa+C2n∆2fa+…+Cnn∆nfa…

Q: flx) = Jx-7 Lim f(3+h)-f(3) ||

A:

Q: Consider the function f defined on the domain -0.5 s x s 0.5 with f(0) = 1, and 0.5 lim h 0 f(x+h)-…

A:

Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Evaluate the limit (1 – cos x)² sin² . lim x2(1 – e)4 COS X - -

A: Given limit is : limx→01-cosx2sin2xx21-ex4  To find the value of the limit.

Q: Zx? +5x-75 Let fl)- X-5 a that f hgs a femovable disqontinutyy at 5 d delermine the value for HS)…

A:

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

A:

Q: f sint dt Evaluate lim

A: We will use DL hospital rule and Newton leibniz formula to solve this question.

Q: c) Consider a function f defined on the real numbers as { 0ifais irrational. S(r) = 1/n ifr= m/n…

A: To show that the function f is continuous at all irrational numbers.

Q: 1 Let f(x) = 1- 1+1-1 Find f(1/2) i. ii. Find the domain of f

A: We have, f(x) = 11-21+11-xi. f12 = 11-21+11-12= 11-21+22-1=11-23=13-23= 3    Answer.....

Q: 1. Explain what it intuitively means for function f(x,y) to be continuous in some domain D.

A: To explain the meaning of the given statement.

Q: Let f be a function satisfying f (x + y) = f (x)f (y)– J4 - f (y) for all x, y eR. and lim f (x) =…

A:

Q: Q3 Suppose f is differentiable at a. Prove lim h→0 f(a+h) — f(a − h) 2h = f'(a).

A: Suppose f is differentiable at a. Prove lim h -> 0 (f(a + h) - f(a - h))/(2h) = f' * (a)

Q: (3) Let f be a positive and continuous function on [О, 1]. Сompute

A: Given: ∫01f(x)f(x)+f(1-x)dx