Q. 15 - Precalculus Enhanced with Graphing Utilities

Question

For what numbers $x, -2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi },$ does the graph of $y= \tan x$ have vertical asymptotes?

Step-by-Step Solution

Answer

The graph of $y=\tan x$ have vertical asymptotes at $-\frac{3\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}, and \frac{3\mathrm{\pi }}{2}$ where $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }.$

Step 1. Given Information

The given function is $y=\tan x.$

We have to find the vertical asymptotes at $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }.$

Step 2. Finding the numbers

For the function, $y=\tan x$ vertical asymptotes occur at odd integer multiples of $\frac{\mathrm{\pi }}{2}.$

So, for $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$ the vertical asymptotes occur at $-\frac{3\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}, and \frac{3\mathrm{\pi }}{2}.$