Find the following limits (a) by investigating values of

Problem 2E In Exercise, choose the best answer for the limit. If a. is ?1. b. does not exist. c. is infinite. d. is 1.

Problem 4E In Exercise, choose the best answer for the limit. If a. is ?. b. is ??. c. does not exist. d. is 1.

Problem 93A Legislative Voting Members of a legislature often must vote repeatedly on the same bill. As time goes on, members may change their votes. Suppose that p0 is the probability that an individual legislator favors an issue before the first roll call vote, and suppose that p is the probability of a change in position from one vote to the next. Then the probability that the legislator will vote “yes” on the n th roll call is given by For example, the chance of a “yes” on the third roll call vote is Source: Mathematics in the Behavioral and Social Sciences. Suppose that there is a chance of p0 = 0.7 that Congressman Stephens will favor the budget appropriation bill before the first roll call, but only a probability of p = 0.2 that he will change his mind on the subsequent vote. Find and interpret the following. a.p2 b.p4 c.p8 d.

Problem 3E In Exercise, choose the best answer for the limit. If , but f(4) does not exist, then a. does not exist. b. is 6. c. is ??. d. is ?.

Problem 1E In Exercise, choose the best answer for the limit. If a. is 5. b. is 6. c. does not exist. d. is infinite.

Problem 5E Decide whether the limit exists. If a limit exists, estimate its value. a. b.

Problem 6E Decide whether the limit exists. If a limit exists, estimate its value. a. b.

Problem 8E Decide whether the limit exists. If a limit exists, estimate its value. a. b.

Problem 9E In Exercise, use the graph to find (i) , (ii) , (iii) , and (iv)f(a) if it exists. a.a = ?2 b.a = ?1

Problem 7E Decide whether the limit exists. If a limit exists, estimate its value. a. b.

Problem 11E Decide whether the limit exists. If a limit exists, find its value.

Problem 10E In Exercise, use the graph to find (i) , (ii) , (iii) , and (iv)f(a) if it exists. a. a = 1 b. a = 2

Problem 12E Decide whether the limit exists. If a limit exists, find its value.

Problem 13E Explain why in Exercise 1 exists, but in Exercise 2 does not. Exercise 1 Decide whether the limit exists. If a limit exists, estimate its value. a. b. Exercise 2 In Exercise, use the graph to find (i) , (ii) , (iii) , and (iv) f(a) if it exists. a.a = ?2 b.a = ?1

Problem 14E In Exercise, why does , even though f(1) = 2? In Exercise, use the graph to find (i) , (ii) , (iii) , and (iv)f(a) if it exists. a. a = 1 b. a = 2

Problem 15E Use the table of values to estimate x 0.9 0.99 0.999 0.9999 1.0001 1.001 1.01 1.1 f(x) 3.9 3.99 3.999 3.9999 4.0001 4.001 4.01 4.1

Problem 16E Complete the table and use the result to find the indicated limits. If f(x) = 2x2 ?4x + 7, find x 0.9 0.99 0.999 1.001 1.01 1.1 f (x) 5.000002 5.000002

Problem 18E Complete the table and use the result to find the indicated limits. If x ?1.1 ?1.01 ?1.001 ?0.999 ?0.99 ?0.9 f (x)

Problem 17E Complete the table and use the result to find the indicated limits. If , find . x 1.9 1.99 1.999 2.001 2.01 2.1 k (x)

Problem 19E Complete the table and use the result to find the indicated limits. If x 0.9 0.99 0.999 1.001 1.01 1.1 h (x)

Problem 20E Complete the table and use the result to find the indicated limits. If x 2.9 2.99 2.999 3.001 3.01 3.1 f (x)

Problem 21E Let . Use the limit rules to find the limit.

Problem 22E Let . Use the limit rules to find the limit.

Problem 23E Let . Use the limit rules to find the limit.

Problem 24E Let . Use the limit rules to find the limit.

Problem 25E Let . Use the limit rules to find the limit.

Problem 26E Let . Use the limit rules to find the limit.

Problem 27E Let . Use the limit rules to find the limit.

Problem 28E Let . Use the limit rules to find the limit.

Problem 29E Let . Use the limit rules to find the limit.

Problem 30E Let . Use the limit rules to find the limit.

Problem 31E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 32E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 33E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 34E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 35E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 36E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 37E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 38E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 39E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 40E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 41E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 42E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 43E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 44E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 45E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 46E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 47E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 48E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 49E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 50E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 51E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 52E Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value.

Problem 54E Use the properties of limits to help decide whether limit exists. If a limit exists, find its value. Let Find .

Problem 55E Use the properties of limits to help decide whether limit exists. If a limit exists, find its value. Let . a. Find . ________________ b. Find .

Problem 56E Use the properties of limits to help decide whether limit exists. If a limit exists, find its value. Let . a. Find . ________________ b. Find .

Problem 57E In Exercise, calculate the limit in the specified exercise, using a table such as in Exercises 1–7. Verify your answer by using a graphing calculator to zoom in on the point on the graph. Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value. Exercise 1 Exercise 2 Use the table of values to estimate x 0.9 0.99 0.999 0.9999 1.0001 1.001 1.01 1.1 f (x) 3.9 3.99 3.999 3.9999 4.0001 4.001 4.01 4.1 Exercise 3 Complete the table and use the result to find the indicated limits. If f(x) = 2x2 ?4x + 7, find x 0.9 0.99 0.999 1.001 1.01 1.1 f (x) 5.000002 5.000002 Exercise 4 Complete the table and use the result to find the indicated limits. If , find . x 1.9 1.99 1.999 2.001 2.01 2.1 k (x) Exercise 5 Complete the table and use the result to find the indicated limits. If x ?1.1 ?1.01 ?1.001 ?0.999 ?0.99 ?0.9 f (x) Exercise 6 Complete the table and use the result to find the indicated limits. If x 0.9 0.99 0.999 1.001 1.01 1.1 h (x) Exercise 7 Complete the table and use the result to find the indicated limits. If x 2.9 2.99 2.999 3.001 3.01 3.1 f (x)

Problem 58E In Exercise, calculate the limit in the specified exercise, using a table such as in Exercises 1–7. Verify your answer by using a graphing calculator to zoom in on the point on the graph. Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value. Exercise 1 Exercise 2 Use the table of values to estimate x 0.9 0.99 0.999 0.9999 1.0001 1.001 1.01 1.1 f(x) 3.9 3.99 3.999 3.9999 4.0001 4.001 4.01 4.1 Exercise 3 Complete the table and use the result to find the indicated limits. If f(x) = 2x2 ?4x + 7, find x 0.9 0.99 0.999 1.001 1.01 1.1 f (x) 5.000002 5.000002 Exercise 4 Complete the table and use the result to find the indicated limits. If , find . x 1.9 1.99 1.999 2.001 2.01 2.1 k (x) Exercise 5 Complete the table and use the result to find the indicated limits. If x ?1.1 ?1.01 ?1.001 ?0.999 ?0.99 ?0.9 f (x) Exercise 6 Complete the table and use the result to find the indicated limits. If x 0.9 0.99 0.999 1.001 1.01 1.1 h (x) Exercise 7 Complete the table and use the result to find the indicated limits. If x 2.9 2.99 2.999 3.001 3.01 3.1 f (x)

Problem 61E Let a. Find . ________________ b. Find the vertical asymptote of the graph of F(x). ________________ c. Compare your answers for parts a and b. What can you conclude?

In Exercise, calculate the limit in the specified exercise, using a table such as in Exercises 1–7. Verify your answer by using a graphing calculator to zoom in on the point on the graph. Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value. Exercise 1 Exercise 2 Use the table of values to estimate . Exercise 3 Complete the table and use the result to find the indicated limits. If f(x) = 2x2 ?4x + 7, find . Exercise 4 Complete the table and use the result to find the indicated limits. If , find . Exercise 5 Complete the table and use the result to find the indicated limits. If . Exercise 6 Complete the table and use the result to find the indicated limits. If . Exercise 7 Complete the table and use the result to find the indicated limits. If .

Problem 62E Let a. Find . ________________ b. Find the vertical asymptote of the graph of G(x). ________________ c. Compare your answers for parts a and b. Are they related? How?

In Exercise, calculate the limit in the specified exercise, using a table such as in Exercises 1–7. Verify your answer by using a graphing calculator to zoom in on the point on the graph. Use the properties of limits to help decide whether the limit exists. If a limit exists, find its value. Exercise 1 Exercise 2 Use the table of values to estimate Exercise 3 Complete the table and use the result to find the indicated limits. If f(x) = 2x2 ? 4x + 7, find Exercise 4 Complete the table and use the result to find the indicated limits. If , find . Exercise 5 Complete the table and use the result to find the indicated limits. If Exercise 6 Complete the table and use the result to find the indicated limits. If Exercise 7 Complete the table and use the result to find the indicated limits. If

Problem 63E How can you tell that the graph in Figure 10 is more representative of the function f(x) = 1/(x ? 1) than the graph in Figure 11? FIGURE 10 FIGURE 11

Problem 64E A friend who is confused about limits wonders why you investigate the value of a function closer and closer to a point, instead of just finding the value of a function at the point. How would you respond?

Problem 65E Use a graph of f(x) = ex to answer the following questions. a. Find . ________________ b. Where does the function ex have a horizontal asymptote?

Problem 66E Use a graphing calculator to answer the following questions. a. From a graph of y = xe?x, what do you think is the value of ? Support this by evaluating the function for several large values of x. ________________ b. Repeat part a, this time using the graph of y = x2 e?x. ________________ c. Based on your results from parts a and b, what do you think is the value of , where nis a positive integer? Support this by experimenting with other positive integers n.

Problem 69E Explain in your own words why the rules for limits at infinity should be true.

Problem 67E Use a graph of f(x) = ln x to answer the following questions. a. Find . ________________ b. Where does the function ln x have a vertical asymptote?

Problem 68E Use a graphing calculator to answer the following questions. a. From a graph of y = x ln x, what do you think is the value of ? Support this by evaluating the function for several small values of x. ________________ b. Repeat part a, this time using the graph of y = x (ln x)2. ________________ c. Based on your results from parts a and b, what do you think is the value of , where n is a positive integer? Support this by experimenting with other positive integers n.

Problem 70E Explain in your own words what Rule 4 for limits means.

Problem 71E Find the following limits (a) by investigating values of the function near the x-value where the limit is taken, and (b) using a graphing calculator to view the function near that value of x.

Problem 72E Find the following limits (a) by investigating values of the function near the x-value where the limit is taken, and (b) using a graphing calculator to view the function near that value of x.

Problem 73E Find the following limits (a) by investigating values of the function near the x-value where the limit is taken, and (b) using a graphing calculator to view the function near that value of x.

Problem 74E Find the following limits (a) by investigating values of the function near the x-value where the limit is taken, and (b) using a graphing calculator to view the function near that value of x.

Problem 75E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 76E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 77E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 78E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 79E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 80E Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x).

Problem 81E Explain why the following rules can be used to find : a. If the degree of p(x) is less than the degree of q(x), the limit is 0. ________________ b. If the degree of p(x) is equal to the degree of q(x), the limit is A/B, where A and B are the leading coefficients of p(x) and q(x), respectively. ________________ c. If the degree of p(x) is greater than the degree of q(x), the limit is ? or ? ?.

Problem 82A APPLY IT Consumer Demand When the price of an essential commodity (such as gasoline) rises rapidly, consumption drops slowly at first. If the price continues to rise, however, a “tipping” point may be reached, at which consumption takes a sudden substantial drop. Suppose the accompanying graph shows the consumption of gasoline, G(t), in millions of gallons, in a certain area. We assume that the price is rising rapidly. Here t is time in months after the price began rising. Use the graph to find the following. a. b. c.G(16) d. The tipping point (in months)

Problem 84A Postage The graph below shows how the postage required to mail a letter in the United States has changed in recent years. Let C(t) be the cost to mail a letter in the year t. Find the following. Source: United States Postal Service. a. ________________ b. ________________ c. ________________ d.

Problem 88A Preferred Stock In business finance, an annuity is a series of equal payments received at equal intervals for a finite period of time. The present value of an n-period annuity takes the form where R is the amount of the periodic payment and i is the fixed interest rate per period. Many corporations raise money by issuing preferred stock. Holders of the preferred stock, called a perpetuity, receive payments that take the form of an annuity in that the amount of the payment never changes. However, normally the payments for preferred stock do not end but theoretically continue forever. Find the limit of this present value equation as n approaches infinity to derive a formula for the present value of a share of preferred stock paying a periodic dividend R. Source: Robert D. Campbell.

Problem 85A Average Cost The cost (in dollars) for manufacturing a particular DVD is C(x) = 15,000 + 6x, where x is the number of DVDs produced. Recall from the previous chapter that the average cost per DVD, denoted by , is found by dividing C(x) by x. Find and interpret .

Problem 86A Average Cost In Chapter 1, we saw that the cost to fly x miles on American Airlines could be approximated by the equation C(x) = 0.0738x + 111.83. Recall from the previous chapter that the average cost per mile, denoted by , is found by dividing C(x) by x. Find and interpret . Source: American Airlines.

Problem 83A Sales Tax Officials in California tend to raise the sales tax in years in which the state faces a budget deficit and then cut the tax when the state has a surplus. The graph below shows the California state sales tax since it was first established in 1933. Let T(x) represent the sales tax per dollar spent in year x. Find the following. Source: California State. a. ________________ b. ________________ c. ________________ d. ________________ e. T(90)

Problem 87A Employee Productivity A company training program has determined that, on the average, a new employee produces P(s) items per day after s days of on-the-job training, where Find and interpret .

Problem 89A Growing Annuities For some annuities encountered in business finance, called growing annuities, the amount of the periodic payment is not constant but grows at a constant periodic rate. Leases with escalation clauses can be examples of growing annuities. The present value of a growing annuity takes the form where R = amount of the next annuity payment, g = expected constant annuity growth rate, i = required periodic return at the time the annuity is evaluated, n = number of periodic payments. A corporation’s common stock may be thought of as a claim on a growing annuity where the annuity is the company’s annual dividend. However, in the case of common stock, these payments have no contractual end but theoretically continue forever. Compute the limit of the expression above as n approaches infinity to derive the Gordon-Shapiro Dividend Model popularly used to estimate the value of common stock. Make the reasonable assumption that i>g. (Hint: What happens to an as n ? ? if 0 <a <1?) Source: Robert D. Campbell.

Problem 90A Alligator Teeth Researchers have developed a mathematical model that can be used to estimate the number of teeth N(t) at time t (days of incubation) for Alligator mississippiensis,where Source: Journal of Theoretical Biology. a. Find N(65), the number of teeth of an alligator that hatched after 65 days. ________________ b. Find and use this value as an estimate of the number of teeth of a newborn alligator. (Hint: See Exercise.) Does this estimate differ significantly from the estimate of part a? Exercise Use a graph of f(x) = ex to answer the following questions. a. Find . ________________ b. Where does the function ex have a horizontal asymptote?

Problem 92A Drug Concentration The concentration of a drug in a patient’s bloodstream h hours after it was injected is given by Find and interpret .

Problem 91A Sediment To develop strategies to manage water quality in polluted lakes, biologists must determine the depths of sediments and the rate of sedimentation. It has been determined that the depth of sediment D(t) (in centimeters) with respect to time (in years before 1990) for Lake Coeur d’Alene, Idaho, can be estimated by the equation D(t) = 155(1 ? e?0.0133t). Source: Mathematics Teacher. a. Find D(20) and interpret. ________________ b. Find and interpret.