Mathematics

Mathematics

Problem

Which of the following lists all xx-values in the interval [0, 2π)[0,\,2\pi) at which the graph of y=csc⁡xy = \csc x has vertical asymptotes? a) x=π2,  3π2x = \dfrac{\pi}{2},\; \dfrac{3\pi}{2} b) x=0,  π,  2πx = 0,\; \pi,\; 2\pi c) x=0,  πx = 0,\; \pi d) x=π2,  π,  3π2x = \dfrac{\pi}{2},\; \pi,\; \dfrac{3\pi}{2}
Intuitive explanation

Vertical asymptotes of csc⁡x\csc x occur wherever the function is undefined—that is, where its reciprocal, sin⁡x\sin x, equals zero. Rather than memorizing asymptote locations, identify all solutions to sin⁡x=0\sin x = 0 within your interval, then verify each one makes csc⁡x\csc x blow up to ±∞\pm\infty. This transforms an abstract question into a concrete zero-finding problem.

Intuitive explanation

Vertical asymptotes of csc⁡x\csc x occur wherever the function is undefined—that is, where its reciprocal, sin⁡x\sin x, equals zero. Rather than memorizing asymptote locations, identify all solutions to sin⁡x=0\sin x = 0 within your interval, then verify each one makes csc⁡x\csc x blow up to ±∞\pm\infty. This transforms an abstract question into a concrete zero-finding problem.

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