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 4QQ

(a)

To determine

To find: The domain and range of the function f(x)=cosxx.

(a)

Expert Solution

Answer to Problem 4QQ

The domain and range of the function f(x)=cosxx are (−∞,0)∪(0,∞) and (−∞,∞) respectively.

Explanation of Solution

Given information: The function is f(x)=cosxx.

Calculation:

The function f(x)=cosxx is defined for all the real values of x except x=0.

So, the domain of the function f(x)=cosxx is set of real numbers except x=0 or the interval (−∞,0)∪(0,∞).

The value of function f(x)=cosxx has no minimum and maximum for all the real values of x.

So, the range of the function f(x)=cosxx is set of real numbers or the interval (−∞,∞).

(b)

To determine

To check: Whether the function f(x)=cosxx is even or odd or neither.

(b)

Expert Solution

Answer to Problem 4QQ

The function f(x)=cosxx is an odd function.

Explanation of Solution

Given information: The function is f(x)=cosxx.

Calculation:

Check the value of f(−x) for the nature of f(x)=cosxx.

Substitute −x for x in the given function.

f(x)=cosxxf(−x)=cos(−x)−x

The trigonometric function cosine is even function. So, cos(−x)=cosx.

Substitute cosx for cos(−x) in the above equation.

f(−x)=cos(−x)−x=cosx−x=−cos(x)x=−f(x)

The function f(x) holds the property f(−x)=−f(x) . So, the given function is an odd function.

Therefore, the function f(x)=cosxx is an odd function.

(c)

To determine

To find: The value of limit limx→∞f(x) for f(x)=cosxx.

(c)

Expert Solution

Answer to Problem 4QQ

The value of limit limx→∞f(x) for f(x)=cosxx is 0.

Explanation of Solution

Given information: The function is f(x)=cosxx.

Calculation:

Evaluate the limit by substitution.

limx→∞f(x)=limx→∞cosxx=cos∞∞=0

Therefore, the value of limit limx→∞f(x) for f(x)=cosxx is 0.

(d)

To determine

To find: The value of limit limx→∞f(x) for f(x)=cosxx with the help of Sandwich Theorem.

(d)

Expert Solution

Answer to Problem 4QQ

The value of limit limx→∞f(x) for f(x)=cosxx is 0.

Explanation of Solution

Given information: The function is f(x)=cosxx.

Calculation:

The value of the trigonometric function cosine lies between and equal to −1 to 1 i.e. −1≤cosx≤1.

Divide each side of inequality by x.

−1≤cosx≤1−1x≤cosxx≤1x

Apply limit to all the sides of inequality.

−1x≤cosxx≤1xlimx→∞(−1x)≤limx→∞cosxx≤limx→∞1x0≤limx→∞cosxx≤0

So, the limit of the function f(x)=cosxx is 0.

Therefore, the value of limit limx→∞f(x) for f(x)=cosxx is 0.

Chapter 2.2, Problem 3QQ

Chapter 2.3, Problem 1QR

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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The domain and range of the function f ( x ) = cos x x .

Concept explainers

Videos

To find: The domain and range of the function f(x)=cosxx.

Answer to Problem 4QQ

Explanation of Solution

To check: Whether the function f(x)=cosxx is even or odd or neither.

Answer to Problem 4QQ

Explanation of Solution

To find: The value of limit limx→∞f(x) for f(x)=cosxx.

Answer to Problem 4QQ

Explanation of Solution

To find: The value of limit limx→∞f(x) for f(x)=cosxx with the help of Sandwich Theorem.

Answer to Problem 4QQ

Explanation of Solution

Chapter 2 Solutions

Additional Math Textbook Solutions