Question

The graph of the function f(x)=tan is given above for the interval x in[0,2 pi] Determine the one-sided limit . Then indicate the equation of the vertical asymptote .

Answer 1

Explanation

We are given the graph below of tan x:

We are required to determine limits and vertical asymptotes of the given limit.

This is achieved thus:

Limit: A limit is a value that the output of a function approaches as the input of the function approaches a given value.

Vertical Asymptote: A vertical asymptote is a vertical line that guides the graph of the function but is not part of it. It can never be crossed by the graph because it occurs at the x-value that is not in the domain of the function.

Therefore, we have:

$\begin{gathered} \lim_{x\to(\frac{\pi}{2})^-}f(x)=\infty \\ \text{ We can deduce from the graph that as the function approaches }\frac{\pi}{2}\text{ from the left,} \\ \text{ the graph approaches positive infinity} \end{gathered}$

This indicates the equation of the vertical asymptote as:

$x=\frac{\pi}{2}$

Also, we have:

$\begin{gathered} \lim_{x\to(\frac{3\pi}{2})^-}f(x)=\infty \\ \text{ We can also deduce that as the function approaches }\frac{3\pi}{2}\text{ from the left, } \\ \text{ the graph approaches positive infinity} \end{gathered}$

This indicates the equation of the vertical asymptote as:

$x=\frac{3\pi}{2}$

Hence, the answers review is:

$\begin{gathered} \lim_{x\to(\frac{\pi}{2})^-}f(x)=\infty \\ Vertical\text{ }asymptote:x=\frac{\pi}{2} \\ \\ \lim_{x\to(\frac{3\pi}{2})^-}f(x)=\infty \\ Vertical\text{ }asymptote:x=\frac{3\pi}{2} \end{gathered}$