Question

consider the behavior of values of sin(1/x) near x = 0. the graph of sin(1/x) shown to the right. answer parts (a), (b), and (c). (a) create a table of values of sin(1/x) for the following values of x. complete the following table. x sin(1/x) π/2 3π/2 5π/2 7π/2 9π/2 11π/2

Explanation:

Step1: Recall sine - function property

We know that the sine function is \(y = \sin(u)\), and here \(u=\frac{1}{x}\). We substitute the given \(x\) - values into \(\frac{1}{x}\) and then find the sine of the result.

Step2: Calculate for \(x = \frac{\pi}{2}\)

First, find \(\frac{1}{x}=\frac{2}{\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{\pi})\approx\sin(0.6366)\approx0.6\).

Step3: Calculate for \(x=\frac{3\pi}{2}\)

Find \(\frac{1}{x}=\frac{2}{3\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{3\pi})\approx\sin(0.2122)\approx0.21\).

Step4: Calculate for \(x = \frac{5\pi}{2}\)

Find \(\frac{1}{x}=\frac{2}{5\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{5\pi})\approx\sin(0.1273)\approx0.13\).

Step5: Calculate for \(x=\frac{7\pi}{2}\)

Find \(\frac{1}{x}=\frac{2}{7\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{7\pi})\approx\sin(0.0909)\approx0.09\).

Step6: Calculate for \(x=\frac{9\pi}{2}\)

Find \(\frac{1}{x}=\frac{2}{9\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{9\pi})\approx\sin(0.0707)\approx0.07\).

Step7: Calculate for \(x=\frac{11\pi}{2}\)

Find \(\frac{1}{x}=\frac{2}{11\pi}\). Then \(\sin(\frac{1}{x})=\sin(\frac{2}{11\pi})\approx\sin(0.0573)\approx0.06\).

Answer:

\(x\)	\(\sin(\frac{1}{x})\)
\(\frac{3\pi}{2}\)	\(\approx0.21\)
\(\frac{5\pi}{2}\)	\(\approx0.13\)
\(\frac{7\pi}{2}\)	\(\approx0.09\)
\(\frac{9\pi}{2}\)	\(\approx0.07\)
\(\frac{11\pi}{2}\)	\(\approx0.06\)