Question

complete the sentence below. the negative value k with the greatest value for which x = k is a vertical asymptote of the graph of y = tan x is (type an exact answer, using π as needed.)

Explanation:

Step1: Recall tangent - asymptote formula

The vertical asymptotes of \(y = \tan x\) are given by \(x=\frac{\pi}{2}+n\pi\), where \(n\in\mathbb{Z}\) (the set of all integers).

Step2: Find negative \(k\) value

We want a negative \(k\) value. Let \(x = k=\frac{\pi}{2}+n\pi\). To get a negative \(k\), we set \(n=- 1\). Then \(k=\frac{\pi}{2}+(-1)\pi=\frac{\pi - 2\pi}{2}=-\frac{\pi}{2}\).

Answer:

\(-\frac{\pi}{2}\)