BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.2, Problem 25E

(a)

To determine

The value of limx0cos2xcosxx2 by graphing and find where it crosses the y-axis.

Expert Solution

Answer to Problem 25E

The value of limx0cos2xcosxx2=1.5.

Explanation of Solution

Using the online graphing calculator and draw the graph of the function f(x)=cos2xcosxx2 as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.2, Problem 25E , additional homework tip  1

Zoom in the graph towards the point where the graph crosses the y-axis as shown below in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.2, Problem 25E , additional homework tip  2

From Figure 2, it is observed that graph of f(x)=cos2xcosxx2 approaches −1.5 as x approaches 0 from either side.

That is, limx0cos2xcosxx2=limx0+cos2xcosxx2=1.5.

Since the right hand limits and the left hand limits are equal, the value of limx0cos2xcosxx2 exists.

Thus, the value of limx0cos2xcosxx2=1.5.

(b)

To determine

The answer obtained in part (a) by evaluating the value of cos2xcosxx2 as x approaches 0.

Expert Solution

Answer to Problem 25E

The value of limx0cos2xcosxx2=1.5.

Explanation of Solution

As x approaches 0, evaluate the function f(x)=cos2xcosxx2 for the numbers 0.5,0.1,0.01,0.001,0.0001,0.5,0.1,0.01,0.001 and 0.0001.

Evaluate the function (correct to 6 decimal places) for the values of x and get the following table of values.

xcos2xcosxx2f(x)=cos2xcosxx2
−0.5−0.3372800.25−1.349121
−0.1−0.0149370.01−1.493758
−0.01−0.0001490.0001−1.499937
−0.001−0.00000150.000001−1.499999
−0.0001−0.0000000150.00000001−1.499999
0.5−0.3372800.25−1.349121
0.1−0.0149370.01−1.493758
0.01−0.0001490.0001−1.499937
0.001−0.00000150.000001−1.499999
0.0001−0.0000000150.00000001−1.499999

Here, it is observed that the value of f(x) gets closer to −1.5 as x approaches to 0 from the either side. That is, limx0cos2xcosxx2=1.5.

Therefore, it is verified from the table that limx0cos2xcosxx2=1.5.

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