Problem 88 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

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 1: 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 \([-1.2,1.2,0.01]\).

Step 2: 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.

Step 3: 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.

Step 4: 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.