Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
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Step 1: Understand the function f(x) = sin(x). The sine function is periodic with a period of 2π, meaning it repeats its values every 2π units along the x-axis.
Step 2: Analyze the range of the sine function. The sine function oscillates between -1 and 1 for all x, so its range is [-1, 1].
Step 3: Determine the end behavior by considering the limits as x approaches positive and negative infinity. Since sin(x) is periodic and bounded, it does not approach a specific value as x approaches infinity or negative infinity.
Step 4: Identify any asymptotes. The sine function does not have any vertical or horizontal asymptotes because it is bounded and periodic.
Step 5: Sketch the graph of f(x) = sin(x). Draw a wave-like pattern oscillating between -1 and 1, repeating every 2π along the x-axis, with no asymptotes.
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