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Chapter 5: Problem 69

If you are given the equation of a cotangent function, how do you find a pair of consecutive asymptotes?

Short Answer

Expert verified
A pair of consecutive asymptotes for a cotangent function can be found at \(x = n\pi\) and \(x = (n + 1)\pi\) where \(n\) is any integer.

Step by step solution

01

Understand the Cotangent Function

Cotangent function, represented as \(cot(x)\), is a periodic function with a period of \(\pi\) radians (or 180 degrees). There are vertical asymptotes at the endpoints of every period.
02

Identify Key Points and Period

For any given cotangent function, you start with the period at zero. Since cotangent function has a period of \(\pi\), the consecutive periods will be at \(n\pi\) where \(n\) is any integer (both positive and negative). The endpoints of these periods, \(n\pi\), correspond to vertical asymptotes.
03

Locate the Asymptotes

With the critical points identified, the asymptotes for a cotangent function will thus be at \(x = n\pi\) for all integer \(n\). Pick successive integers to find a pair of consecutive asymptotes. For example, for consecutive integers 1 and 2, the consecutive asymptotes will be at \(\pi\) and \(2\pi\) respectively.

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