math233

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MATH233 Unit 1: Limits

Individual Project Assignment: Version 2A

IMPORTANT: Please see Part b of Problem 5 below for special directions. This is mandatory.

Note: All work must be shown and explained to receive full credit.

1. Using a graphing utility from the Internet or Excel, graph the following functions. Based on the

graphs, estimate the given limit. Make sure to include the graphs in your answer form, and

explain how you found your limit estimates.

a. lim𝑥 →0 100

50𝑥+1

b. lim𝑥 →∞ 𝑥2+1 𝑥2

2. Find the limit (if it exists) of the following functions by completing the given tables. Round your

answers to the nearest ten-thousandths.

a. Let F(x) = x + 1. Find lim𝑥 →1F(𝑥).

x 0.9 0.99 0.999 1 1.001 1.01 1.1 F(x)

b. Let G(x) = 5 (𝑥 −2)2

. Find lim𝑥 →2G(𝑥).

x 1.9 1.99 1.999 2 2.001 2.01 2.1 G(x)

3. Answer the following questions thoroughly based on the given graph of f(x).

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a. Is f(x) continuous at x = −1?

b. Is f(x) continuous at x = 2?

c. Is f(x) continuous at x = 4?

4. Let 𝐴(𝑛) = (1 + 𝑛) 1 𝑛. The limit of this function as n approaches 0 is a value that is very

useful in some business applications.

a. Complete the table below by calculating A(n), using the given values of n. Round

your answer to the nearest ten-thousandths.

n -0.1 -0.01 -0.001 -0.0001 .0001 0.001 0.01 0.1 A(n)

b. Based on the table, estimate the following values:

i. lim𝑛→0−𝐴(𝑛)

ii. lim𝑛→0+𝐴(𝑛)

iii. lim𝑛→0 𝐴(𝑛)

5. The cost, C (in millions of dollars) for a software company to seize x% of an illegal version of

a gaming software that they developed is modeled by the following function:

𝐶(𝑥) = 𝑀𝑥 50−0.5𝑥

0 ≤ 𝑥 < 100

a. Choose a value of M between 20 and 120 for this function.

b. Important: By Wednesday night at midnight, submit a Word document stating only your name and your chosen value for M in Part a. Submit this in the Unit 1

IP submissions area. This submitted Word document will be used to determine

the Last Day of Attendance for government reporting purposes.

c. Find the cost of seizing 50%, 60%, 70%, 80%, and 90% of the illegal software.

d. Find the lim𝑥→100−𝐶(𝑥). Explain briefly what this limit means in terms of the given

scenario.

6. A startup company invested $30,000 for the research and development of a new hardware

plus an additional $80 expense for each unit produced. The total cost is then modeled by the

function 𝐶(𝑥) = 80𝑥 + 30,000, where x is the number of units produced.

a. Find the average cost function, A(x), that models the average cost per unit of the

hardware. (Use the Internet to research the formula for the average cost function.)

b. Find the average cost per unit if 1,000 units, 10,000 units, and 100,000 units of the

hardware are produced.

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c. What is the limit of the average cost as the number of units produced increases?

7. Which intellipath Learning Nodes helped you with this assignment?