Chapter 12.3, Problem 54E

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Please help me organize the proof of the following theorem:

How come that I marked ?

In Exercises 1-14, state whether each statement is true or false. If false, give a reason. 1. The set of stores located in the state of Wyoming is a well- defined set. 2. The set of the three best songs is a well-defined set. 3. maple = {oak, elm, maple, sycamore} 4{} cơ 5. {3, 6, 9, 12,...} and {2, 4, 6, 8, ...} are disjoint sets. 6. {Mercury, Venus, Earth, Mars} is an example of a set in roster form. 7. {candle, picture, lamp} = {picture, chair, lamp } 8. {apple, orange, banana, pear} is equivalent to {tomato, corn, spinach, radish}.

Exercises Evaluate the following limits. 1. lim cot x/ln x +01x 2. lim x² In x +014 3. lim x* x0+ 4. lim (cos√√x)1/x +014 5. lim x2/(1-cos x) x10 6. lim e*/* 818 7. lim (secx - tan x) x-x/2- 8. lim [1+(3/x)]* x→∞0

iation 7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x) xia exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A similar result holds for b. x-a 8. In reference to Corollary 5.4, give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1] but whose derivative is not bounded there. 9. Recall that a fixed point of a function f is a point c such that f(c) = c. (a) Show that if f is differentiable on R and f'(x)| x if x 1 and hence In(1+x) 0. 12. For 0 л/2. (Thus, as x л/2 from the left, cos x is never large enough for x+cosx to be greater than л/2 and cot x is never small enough for x + cot x to be less than x/2.)

1. Show that f(x) = x3 is not uniformly continuous on R. 2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00). 3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2]. 4. Show that f(x) = mx + b is uniformly continuous on R. 5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on (0, 1]. 6. Show that if f is uniformly continuous on [a, b] and uniformly continuous on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous on [a, b]U D. 7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6 to conclude that f is uniformly continuous on [0, ∞). 8. Show that if D is bounded and f is uniformly continuous on D, then fis bounded on D. 9. Let f and g be uniformly continuous on D. Show that f+g is uniformly continuous on D. Show, by example, that fg need not be uniformly con- tinuous on D. 10. Complete the proof of Theorem 4.7. 11. Give an example of a continuous function on Q that cannot be continuously extended to R. 12.…

can I see the steps for how you got the same answers already provided for μ1->μ4. this is  a homework that provide you answers for question after attempting it three tries