6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn converges pointwise and determine the limit. (ii) Use the inequality | sin a| < |x\, x eR to show that fn converges uniformly to the above function in (0, 0). (iii) Calculate the limit na4 – sin(nx) lim dx. n 00 /1 n2x Justify any possible exchange of the integral with the limit.

Q: Consider the following functions: f(x)=x√3-x; g(x) = x³ - 6x² + 3; h(x)= = ²-9 I² + 1 a) Use the…

A:

Q: 3. Use a limit to find f'(4) for f(x) = x²-x.

A:

Q: 1. Let f(x) = x3 – 3a² – 9x + 10 for 1< æ < 9. (a) Use f'(x) to determine where f is increasing and…

A:

Q: 2 2. Is f(x) = { = { x²-1 x+1 if x = -1 if x −1 continuous at x = −1? (Show your work using limits.)

A: See this solution in below.

Q: Consider the function f(x) = - +7. In this question you will calculate dx by using the definition…

A: The Riemann sum defined as,                                       S =∑i=1∞fxi∆x

Q: Which statement is true about a function f(x), where f'(x) = 3x²_²? X Of is increasing for x > 0.874…

A: Given here the function and put here

Q: The sequence of functions {f,(x)} where fn(x) = x" converges on the interval [0, 1] to a function…

A:

Q: 1. Function f is continuous on (-00,00). Find constant k. f(x) [kx² + 2x x-kx if x ≥ 2 if x < 2

A:

Q: (III) If f'(x) > 0 for every x, then f(z) is an increasing function (in fact, f(z) is strictly…

A:

Q: 7.A) Estimate the value of f'(1) by substituting small values for h in the difference quotient…

A:

Q: Why does the limit as x goes to 0 of cos(1/x) not exist? Can use two different sequences:…

A: Solution

Q: Consider the function f(x) = *+7. 4 In this problem you will calculate | + 7) dx by using the…

A:

Q: Which of the following functions are continuous on the given interval? Assume k is a positive…

A:

Q: Find the limit

A:

Q: H(3)=6 and the limit as x approaches 3 of H(x) is 6. Mustache be continuous at (3,6)? State why.

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.