Understanding the Cotangent Function and Its Asymptotic Behavior

Dec 03, 2025

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In this analysis, we explore the graph of y = cot(x) as x approaches 0 from the positive side. As x > 0 nears 0, the values of cot(x) rise infinitely, reflecting that the graph possesses a vertical asymptote along the upper y-axis. This demonstrates that, similar to previous examples, the two-sided limit of cot(x) as x approaches 0 does not exist. We will delve deeper into the properties of cotangent, evaluating its behavior in various limits and graphing its characteristics.

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Understanding the Cotangent Function and Its Asymptotic Behavior

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Example 3Evaluate SolutionConsider the graph of y= cot x below. As x > 0 gets closer and closer to 0, the values of cot x get larger and larger. Equivalently, the graph of y = cot x, for x > 0, has the upper y-axis as a vertical asymptote. Thus Observe that, as in Example 2, the two-sided limit does not exist.