Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Question

Chapter 2.2, Problem 4E

(a)

To determine

To find: The value of limx→∞f(x) for f(x)=3x3−x+1x+3 with the help of graph and table.

(a)

Expert Solution

Answer to Problem 4E

The value of limx→∞f(x) for f(x)=3x3−x+1x+3 is ∞ .

Explanation of Solution

Given information:The function is f(x)=3x3−x+1x+3 .

Calculation:

Find the limit graphically.To graph a function y=3x3−x+1x+3 , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press Y= key and enter right hand side of the equation y=3x3−x+1x+3 after the symbol Y1 . Enter the keystrokes (3X^3−X+1)÷(X+3) .

The display will show the equation,

Y1=(3X^3−X+1)/(X+3)

Press the window key and adjust the window to [−10,10] by [−100,300] .Now, press the GRAPH key and TRACE key to produce the graph of given function in standard window as shown below.

Calculus: Graphical, Numerical, Algebraic, Chapter 2.2, Problem 4E , additional homework tip 1

Figure (1)

As observed from graph, the function f(x)=3x3−x+1x+3 tends to ∞ as x tends to ∞ .

Find the limit by table.To make the table of function y=3x3−x+1x+3 , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press Y= key and enter right hand side of the equation y=3x3−x+1x+3 after the symbol Y1 . Enter the keystrokes (3X^3−X+1)÷(X+3) .

First set the Table setup, enter the keystrokes 2ndWINDOW , then set to independent variable to ask.Now draw the table, enter the keystrokes 2ndGRAPH , and then enter the values of X to 100 , 200 , 300 , 400 , −100 , −200 and −300 . The table is shown in window.

Calculus: Graphical, Numerical, Algebraic, Chapter 2.2, Problem 4E , additional homework tip 2

Figure (2)

As observed from table the value of Y approaches to 0 as x approaches to infinity.

Therefore, the value of limx→∞f(x) for f(x)=3x3−x+1x+3 is ∞ .

(b)

To determine

To find:The value of limx→−∞f(x) for f(x)=3x3−x+1x+3 with the help of graph and table.

(b)

Expert Solution

Answer to Problem 4E

The value of limx→−∞f(x) for f(x)=3x3−x+1x+3 is ∞ .

Explanation of Solution

Given information:The function is f(x)=3x3−x+1x+3 .

Calculation:

To find the limit of the function f(x)=3x3−x+1x+3 with the help of the graph as x tends to −∞ , consider Figure (1) drawn in part (a). From the graph it can be observed that the function tends to ∞ as x tends to −∞ .

To find the limit of the function f(x)=cos(1x) with the help of the table as x tends to −∞ , consider Figure (2) drawn in part (a). From the table it can be observed that the function approaches to ∞ as x approaches to a big value on the left side of number line or to −∞ .

Therefore, the value of limx→−∞f(x) for f(x)=3x3−x+1x+3 is ∞ .

(c)

To determine

To find:The horizontal asymptote to the function f(x)=3x3−x+1x+3 .

(c)

Expert Solution

Answer to Problem 4E

There is no horizontal asymptote to the function f(x)=3x3−x+1x+3 .

Explanation of Solution

Given information:The function is f(x)=3x3−x+1x+3 .

Calculation:

The horizontal asymptote of the graph of a function y=f(x) is the line y=a if limx→∞f(x)=a or limx→−∞f(x)=a .

From part (a) and (b), it can be observed that the value of both left hand and right hand limit of the function is not any real number. So, horizontal asymptote does not exist.

Therefore, there is no horizontal asymptote to the function f(x)=3x3−x+1x+3 .

Chapter 2.2, Problem 3E

Chapter 2.2, Problem 5E

Chapter 2 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 2.1 - Prob. 1QR Ch. 2.1 - Prob. 2QR Ch. 2.1 - Prob. 3QR Ch. 2.1 - Prob. 4QR Ch. 2.1 - Prob. 5QR Ch. 2.1 - Prob. 6QR Ch. 2.1 - Prob. 7QR Ch. 2.1 - Prob. 8QR Ch. 2.1 - Prob. 9QR Ch. 2.1 - Prob. 10QR

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.1 - Prob. 43E Ch. 2.1 - Prob. 44E Ch. 2.1 - Prob. 45E Ch. 2.1 - Prob. 46E Ch. 2.1 - Prob. 47E Ch. 2.1 - Prob. 48E Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - 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