5 Give an example of a sequence of continuous real-valued functions f₁, f2,... on [0, 1] and a continuous real-valued function f on [0, 1] such that for each x = [0, 1] but f(x) = lim f(x) k→∞o [ ²f = lim k→∞0 • Sot fik.

Suppose that f(x) is a continuous function on the interval[ – 3, 1] with f( – 3) = - 9 and f(1) = 9. Determine which choice best describes the following statement. "f(x) = 0 for some r in the interval [-3, 1]" O Always false Always true Sometimes true and sometimes false

Give an example of a function that is continuous on (-∞,0) U (0, 1] U [1,0). Use limits and theory from class to explain why your function meets the criteria.

Describe the interval(s) on which the function is continuous. (TX) 4 (−3, −1), (−1, 1), (1, 3), (3, 5), ... f(x) = sec O O (-∞0, ∞0) … ..., (−6, −2), (−2, 2), (2, 6), (6, 10), ... O ..., (-2π, -π), (−ï, 0), (0, π), (π, 2π), ... 3π 3π π ○ ... (-³7, - 7), (-7, 7-) (½, ³7), (³7, 57), ... 2 22. 2 2

Consider the graph below. (-1,4) (1, 4) 1, 52) (3, 4) (-4, 5) Determine if f is continuous at x = c. (a) x = 4 (b) x = 1 (c) x = -1 (d) x = 3 (e) x = 5

Suppose that f(x) is a continuous function on the interval [0, 4] with f(0) = -10 and f(4) = 10. Determine which choice best describes the following statement. “f(x) = 0 for some x in the interval [0, 4] I. Sometimes true and sometimes false II. Always false III. Always true

Suppose the function f is continuous and has domain (-∞,00) and has all of the following characteristics. Use the information given to answer the questions that follow. f(0) = ft(5) = 0 f'(x) > 0 on the intervals (-o, 0) and (5, ∞) ft(x) 0 on the interval (1, co) f"(x) < 0 on the interval (-o, 1) a. On what interval(s) is f increasing? а. b. For what value(s) of x does f have a local maximum? b. c. On what interval(s) is ƒ concave up? с. d. For what values(s) of x does f have any inflection point(s)? d. e. Use the information to sketch the graph of f.

Find the limit lim h→0 lim h→0 f(-9+h)- f(-9) h f(−9+ h) – f(−9) h = if f(x) = 829 - 6x². I

Characterize the discontinuity for F(X)

10. Consider a continuous function f with the properties that f is concave up on the interval [-1,3] and concave down on the interval [3, 5]. Which of the following statements is true? cancave up (A) f"(2) >0 and f" (4) 0. (E) Both (B) and (C) (C)/f" (3) > 0 and x = 3 is a point of inflection of f.