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- 4. Find Lim?→0^+ (1/(e^x - 1) - 1/x)Let h(x) = (x2 - 2x - 3)/(x2 - 4x + 3). a. Make a table of the values of h at x = 2.9, 2.99, 2.999, and so on. Then estimate limx-->3 h(x). What estimate do you arrive at if you evaluate h at x = 3.1, 3.01, 3.001,......instead? b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x--> 3.Let ƒ(x) = (x2 - 1)/( | x| - 1). Make tables of the values of ƒ at values of x that approach c = -1 from above and below. Then estimate limx→ -1 ƒ(x).
- (b) If we keep the first part of the hypothesis of Theorem 5.3.6(L’Hospital’s Rule) the same but we assume that lim f'(x)/g'(x) = ∞, x→a does it necessarily follow that lim f(x)/g(x)= ∞?, x→aNote: You may need to assume the fact that lim M→+∞ Mne−M = 0 for all n.Sales of the text Calculus and You have been declining continuously at a rate of 2% per year. Assuming that Calculus and You currently sells 4,700 copies per year and that sales will continue this pattern of decline, calculate total future sales of the text. HINT [Use a model of the form Aert.]What is the long term behavior of your hand drawn solution; that is, what is lim t-> infinity M(t) equal to?
- find the limit: limx→∞ (ln x)^3 / x^2 find the point(s) on the hyperbola y^2-x^2=4 that is/are closest to the point (2,0)(3) Using the appropriate approach covered in Unit 6 of this Course along with knowledgegained in Unit 1 and 2, show that: (a) Limx→0 (1/x-1 +1/x+1)/x = -2 (b) Limh→0 (h/√5h + 4 − 2) = 4/5(c) Limt→1(t3- 1/t-1) = 3Let ƒ(x) = (x^2 - 9)/(x + 3). a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001, and so on as far as your calculator can go. Then estimate limxS -3 ƒ(x). What estimate do you arrive at if you evaluate ƒ at x = -2.9, -2.99, -2.999,c instead? b. Support your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as xS -3. c. Find limxS -3 ƒ(x) algebraically.
- Find the slope of the tangent line to the parabola y=4x-x^2 at the point (1,3) 1) using the definition: m=lim->a f(x)-f(a)/x-a 2)using the equation: m=lim->0 f(a+h)-f(a)/hLet ƒ(x) = (x2 - 9)/(x + 3). a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001, and so on as far as your calculator can go. Then estimate limx--> -3 ƒ(x). What estimate do you arrive at if you evaluate ƒ at x = -2.9, -2.99, -2.999,...... instead? b. Support your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as x -->-3.Given lim (4x-3) as x approaches 1. Use the definition of a limt to find a number delta such that the absolute value of x-a is less than delta when the absolute value of f(x)-L is less than 0.08