Finding a symmetric interval The function f in the figure satisfies s lim f1x2 = 5. For each value of e, find all values of xS4 d 7 O such that |f1x2 - 51 6 e whenever 0 6 lx - 4| 6 d. a. e = 2 c. For any e 7 0, make a conjecture about the corresponding val- ues of d satisfying (3). b. e = 1 y A 7 6. 5+ 4 1+ 1 3 4 5 6 7 X 2. 3. 2.
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- a). Determine the interval(s) on which f(x) =4x−12x−1is continuous. b). Find lim x→0 4x−12x−1analytically (use algebra).(a) for g(t)= t-9/square root t-3 . Make table of values with at least six appropriate inputs to evaluate the limits as t tends to 9. (b) make a conjecture about the value of lim t-9/square root t-3. t->9A function is a ratio of quadratic functions and has a verticalasymptote x =4 and just one -intercept, x =1 . It isknown f that has a removable discontinuity at x =- 1 and lim x->-1f(x) = 2. Evaluate a) f (0) b) limx->∞f(x)
- 1- give an example of a function with two zeros (f(c)=0), one of which cannot be found by the bisection method (you may sketch a graph.) 2- find the constant c so that the lim_{x->3} (x^2+x+c)/(x^2-5x+6) exists. 3- determine at which points, the function f(x)=x2 if x∈Q if x and f(x)=-x2 if x∈R/Q is continuous.2. a) Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove that g(x) = −f(x) for all x ∈ R. b) Suppose neither limx→0 f(x) nor limx→0 g(x) exists. Show that limx→0 [f(x)g(x)] may exist. c) Suppose f is differentiable everywhere and f'(x) > 0 for all numbers x except for a single number d. Prove that f is always strictly increasing. d) Consider the function f(x) = 4√ x on some closed interval and by applying the Mean Value Theorem, show that3 <4√82 <325/108 e) Prove that limx→4 √x = 2 using the ε-δ-definition. Hint: √x − 2 =x − 4/√x + 2Sketch a graph of the function f that has the following properties: f is continuous on R\{−1, 4}, f has a vertical asymptote x = −1, lim x→4 f(x)=−2, f has NO horizontal and oblique asymptotes, and f has the following table of signs for f, f′ and f′′ f f' f'' (-infinity,-2) - + -2 1 0 0 (-2,-1) + + -1 dne dne dne (-1,1) + - 1 0 dne dne (1,4) - + 4 1 dne dne (4,infinity) - -
- Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)=e^1.9x−e^3.1x/x. We want to find the limit limx→0 e^1.9x−e^3.1x/x.Start by calculating the values of the function for the inputs listed in this table. x x f(x)f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears limx→0 e^1.9x−e^3.1x/x=Let E ⊂ R, c a limit point of E, and f, a real-valued function with domain E. Let k ∈ R. Provethat if there exists a δ > 0 such that f(x) > k for all x ∈ E with 0 < |x − c| < δ, and if limx→cf(x) exists, thenlimx→cf(x) ≥ k.which of the following statements is true about the function 3x-2, if x<2 r(x)= -8 if x=-2 5x+4 if x>2 a. f(x) is not continuos because f(-2) is not defined b. f(x) s not continuos at x=-2 because lim f(x)≠ f(-2) x->-2 c. f(x) is not continuous at x=-2 only d. f(x) is not continuos at x=-2 because lim f(x) ≠ lim f(x) x->-2- x->-2+
- 4 Given f(x), g(x) are both continuous at x = 2, f(2) = 5, andg(2) = 3; evaluate limx→2 3f(x)/g(x)Determine whether f is differentiable at x=0 by considering limh→0f(0+h)−f(0) / h. f(x)=7−x Choose the correct answer below. A. The function f is differentiable at x=0 because both the left- and right-hand limits of the difference quotient exist at x=0. B. The function f is not differentiable at x=0 because the left- and right-hand limits of the difference quotient do not exist at x=0. C. The function f is differentiable at x=0 because the graph has a sharp corner at x=0. D. The function f is not differentiable at x=0 because the left- and right-hand limits of the difference quotient exist at x=0, but are not equal.Let {fn} be a sequence of nonnegative measurable functions and fn (x) → f (x) on a set A of finite measure and gn (x) = min (g (x), fn (x)), where is a bounded measurable which vanishes outside a set A. Moreover, g < f. a) Give a rigorous proof (using e and Ne notations) of fact that gn (x) → g (x) for each x E A.b) Does a) allow us to conclude that lim fp 9n = Se g? If it is not the case, explain why?