Which of the following is false? a. If f(x) is a real function defined in a neighborhood of 0 and if limx→0 f(x) exists, then f is continuous at 0. b. If (an) and (bn) are two increasing sequences, each of positive terms, then so is the sequence (a,bn). C. The function f(x) = e* cos(x)(x* + 7)- is continuous on the whole set R. d. Let an E (-∞, 10) for all n E N. If (an) is an increasing sequence, then it converges. e. Every Cauchy sequence is bounded.

Q: Suppose that ƒ′(x)<= 1 for 1<= x<= 4. Show that ƒ(4) - ƒ(1)<=3.

A: Show that:

Q: B. The graph of f(x) is given by (-14) (1,4) (-1,-2) (5,-2) (3, -4) (4, 5) Determine is f is…

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Q: q). State the equations of the horizontal asymptotes. r). State the equations of the vertical…

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Q: Define the sequence of functions {fn(x)} on [0, 1] by { 2n2x; 2n(1 — пх); 0; fn(x) = 2n 1 n (a)…

A: Given: fn(x)=2n2x;            x≤12n2n(1-nx);    12n<x ≤1n0;              1n<x, x∈[0, 1].

Q: Suppose that the increasing function ƒ(x) is smooth for x >= 0 and that ƒ(0) = a. Let s(x) denote…

A: Given information:

Q: When f(x)→±∞ as x→a from the left or the right, x = a is a ________ ________ of the graph of f.

A: Given that when f(x)→±∞ as x→a . We need determine the nature of x as x=a such that f(x)→±∞.

Q: 1. Let f(x) = r3 – 12r. This function is: • Increasing on (-0, -2) and on (2, 00) • Decreasing on…

A: Solve by finding minima and maxima

Q: 1.2. For the following functions 1.2.1 (a) Find the domain. x + 13 1- x2 + 4x + 3 f(x) = (b) Discuss…

A: Domain set of all input values where function defined

Q: 6. Suppose f and g are functions with domain R. If bothf and g are even but f+g is odd, then prove…

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Q: 6. Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove…

A: A function defined on a domain R is said to be even if , f(-x)=f(x) ∀x∈R A function defined on  a…

Q: 6. For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→Z a(x) = B(n)…

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Q: Consider the function f(x) given by the equation. f(x) = 2023 + 2022 + 2023 2021 + 2022 2021 2020 +…

A: We have to find the equation of the line tangent to the graph of f(x) at the point with x-coordinate…

Q: Let 2x - x if 0 4 (a) For each of the numbers 2, 3, and 4, dctcrmine whether g is continuous from…

A: We are given                 . For (a), For each numbers 2, 3, and 4, we need to determine whether g…

Q: Answer true or false. If f is not piecewise continuous on [0, ∞o>), then L{f(t)} will not exist.…

A:

Q: Decide if the statement is true or false, and select the correct explanation. Ifa < b andf'(x) is…

A: Function f is increasing on I if for every a &lt;b in I , f(a)≤f(b) Where I denote integer number.

Q: Suppose that 0 &lt;ƒ′(x) &lt; 1/2 for all x-values. Show that ƒ(-1) &lt; ƒ(1) &lt; 2 + ƒ(-1)

A: From the given inequality the behaviour of the function can be determined as.

Q: Show that f(x) has a removable discontinuity at x = 0 and determine what value for f(0) wou make…

A: Given that ,  fx=9x+-8x+9xx-1if x≠0 and x≠13if x=0 We know that, A function fx is said to have…

Q: b) Discuss the continuity conditions of the function- f(x) which shown in figure at x= 0,x = 1,x= 2…

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Q: f = 0(g) implies g = (f).

A:

Q: Q5 Prove that the function g(x) = sin(1/x) for x 0 and g(0) = 0 is discontinuous at = 0. You can…

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