# To estimate: The value of lim x → 0 sin x sin π x by graphing the function f ( x ) = sin x sin π x . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 2.2, Problem 26E

(a)

To determine

## To estimate: The value of limx→0sinxsinπx by graphing the function f(x)=sinxsinπx.

Expert Solution

The value of limx0sinxsinπx0.32.

### Explanation of Solution

Using the graphing calculator, the graph of the function f(x)=sinxsinπx is drawn and shown below in Figure 1. Zoom the graph towards the point where the graph crosses the y-axis as shown below in Figure 2. From Figure 2, it is observed that graph of f(x)=sinxsinπx approaches 0.32 as x approaches 0 from either side.

Since the right hand and the left hand limits are equal, the value of limx0sinxsinπx exists.

That is, limx0sinxsinπx=limx0+sinxsinπx0.32.

Thus, the value of limx0sinxsinπx0.32.

(b)

To determine

Expert Solution

### Explanation of Solution

As x approaches 0, evaluate the function f(x)=sinxsinπx for the x values 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.

 x sinx sinπx f(x)=sinxsinπx −0.5 −1.571428571 −0.9999998 0.318346 −0.1 −0.314285714 −0.309137252 0.318311 −0.01 −0.031428571 −0.031423398 0.318228 −0.001 −0.003142857 −0.003142852 0.318309 −0.0001 −0.000314286 −0.000314286 0.318309 0.5 1.571428571 0.9999998 0.318345 0.1 0.314285714 0.309137252 0.318311 0.01 0.031428571 0.031423398 0.318309 0.001 0.003142857 0.003142852 0.318309 0.0001 0.000314286 0.000314286 0.318309

From the above table, it is observed that the value of f(x) is approximately closer to 0.32 as x approaches to 0 from either side. That is, limx0sinxsinπx0.32.

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