Question

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

Answer 1

From the given graph, you can conclude that as

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

from the right

$f(x)\rightarrow-\infty.$

Therefore:

$\lim_{x\to(\frac{\pi}{2})^+}f(x)=-\infty.$

Also, as

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

from the right

$f(x)\rightarrow\infty.$

Therefore:

$\lim_{x\to(\frac{3\pi}{2})^+}f(x)=\infty.$

Both limits and the graph lead you to conclude that, there are two vertical asymptotes with equations:

$\begin{gathered} x=\frac{\pi}{2}, \\ x=\frac{3\pi}{2}. \end{gathered}$

Answer:

$\begin{gathered} \lim_{x\to(\frac{\pi}{2})^+}f(x)=-\infty, \\ x=\frac{\pi}{2}. \\ \\ \lim_{x\to(\frac{3\pi}{2})^+}f(x)=\infty, \\ x=\frac{3\pi}{2}. \end{gathered}$