Chapter 2.2, Problem 26E

(a)

To determine

To estimate: The value of $\underset{x\to 0}{\lim }\frac{\sin x}{\sin \pi x}$ by graphing the function $f\left(x\right)=\frac{\sin x}{\sin \pi x}$.

Answer to Problem 26E

The value of $\underset{x\to 0}{\lim }\frac{\sin x}{\sin \pi x}\approx 0.32$.

Explanation of Solution

Using the graphing calculator, the graph of the function $f\left(x\right)=\frac{\sin x}{\sin \pi x}$ 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 $f\left(x\right)=\frac{\sin x}{\sin \pi x}$ approaches 0.32 as x approaches 0 from either side.

Since the right hand and the left hand limits are equal, the value of $\underset{x\to 0}{\lim }\frac{\sin x}{\sin \pi x}$ exists.

That is, $\underset{x\to 0^{-}}{\lim }\frac{\sin x}{\sin \pi x}=\underset{x\to 0^{+}}{\lim }\frac{\sin x}{\sin \pi x}\approx 0.32$.

Thus, the value of $\underset{x\to 0}{\lim }\frac{\sin x}{\sin \pi x}\approx 0.32$.

(b)

To determine

To check: The correctness of the answer obtained in part (a) by evaluating the value of $f\left(x\right)=\frac{\sin x}{\sin \pi x}$ as x approaches 0.

Explanation of Solution

As x approaches 0, evaluate the function $f\left(x\right)=\frac{\sin x}{\sin \pi x}$ for the x values $-0.5,-0.1,$$-0.01,-0.001,-0.0001,0.5,0.1,0.01,0.001\text{ and }0.0001$.

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

$x$	$\sin x$	$\sin \pi x$	$f\left(x\right)=\frac{\sin x}{\sin \pi x}$
−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 $f\left(x\right)$ is approximately closer to 0.32 as x approaches to 0 from either side. That is, $\underset{x\to 0}{\lim }\frac{\sin x}{\sin \pi x}\approx 0.32$.