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Chapter 5: Problem 88

Graph \(y=\sin \frac{1}{x}\) in a \([-0.2,0.2,0.01]\) by \([-1.2,1.2,0.01]\) viewing rectangle. What is happening as \(x\) approaches 0 from the left or the right? Explain this behavior.

Short Answer

Expert verified
As \(x\) approaches 0, from either the left or right, the function \(y=\sin \frac{1}{x}\) oscillates between -1 and 1 increasingly quickly. This is because the \( \frac{1}{x}\) part makes the frequency of oscillation very high. So, as \(x\) gets closer to zero, the frequency gets higher, and the graph begins to appear as a thick vertical band due to the increased oscillation speed.

Step by step solution

01

Define the Function

The function to be visualized is \(y=\sin \frac{1}{x}\). This is a sinusoidal function with a period dependent on \(x\), and it is required to graph this function in the viewing rectangle \([-0.2,0.2,0.01]\) by \([-1.2,1.2,0.01]\).
02

Graphing

Using a graph plotting tool or software, plot the function on the required viewing rectangle. Observe the function's behavior for various small values of \(x\) approaching 0.
03

Observing the Behavior as \(x\) Approaches 0

When \(x\) is near 0, the function oscillates more and more rapidly between -1 and 1 because the frequency \( \frac{1}{x}\) is very high. On the graph, it is observed as a thick band due to the high-frequency oscillations.
04

Interpreting the Behavior

As \(x\) approaches 0 from the left or the right, the value of \(y=\sin \frac{1}{x}\) oscillates increasingly fast between -1 and 1 because \( \frac{1}{x}\) becomes a very large value. This results in the sine function oscillating rapidly.

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