
Try these problems.

Consider Find all vertical asymptotes.
There are vertical asymptotes There are no vertical asymptotes
Consider Find all vertical asymptotes.
There are vertical asymptotes There are no vertical asymptotes

Give intervals on which each of the following functions are continuous. Write combinations of intervals going from left to right on the number line.

(a)
$\frac {1}{e^x+1}$ is continuous on $(\answer {-\infty },\answer {\infty })$.
(b)
$\frac {1}{x^2-1}$ is continuous on $(\answer {-\infty },\answer {-1})$ and $(\answer {-1},\answer {1})$ and $(\answer {1},\answer {\infty })$.
(c)
$\sqrt {5-x}$ is continuous on $(\answer {-\infty },\answer {5})$.
(d)
$\sqrt {5-x^2}$ is continuous on $(\answer {-\sqrt {5}},\answer {\sqrt {5}})$.

Let Is $f$ continuous everywhere?

Yes No

Let

Find

(a)
$\lim _{x\to 0^{-}} g(x)\begin {prompt} = \answer {-2}\end {prompt}$
(b)
$\lim _{x\to 0^+} g(x)\begin {prompt} = \answer {-2}\end {prompt}$
(c)
$\lim _{x\to 3^{-}} g(x)\begin {prompt} = \answer {-\infty }\end {prompt}$
(d)
$\lim _{x\to 3^{+}} g(x)\begin {prompt} = \answer {\infty }\end {prompt}$
(e)
$\lim _{x\to -\infty } g(x)\begin {prompt} = \answer {0}\end {prompt}$
(f)
$\lim _{x\to +\infty } g(x)\begin {prompt} = \answer {1}\end {prompt}$
The Intermediate Value Theorem states: If $f$ is a continuous function for all $x$ in the closed interval $[a,b]$ and $r$ is between af(a) and bf(b) , then there is a number uf(u) in $[a, b]$ such that f(u) = rf(r) = u .
Consider the following graph:

Is the function continuous at $x=0$ or $x=8$?

$f$ is continuous at both $x=0$ and $x=8$. $f$ is continuous at $x=0$ but not at $x=8$. $f$ is continuous at at $x=8$ but not at $x=0$. $f$ is not continuous at $x=0$ and $x=8$.

Let $f$ be continuous on $\left [1,5\right ]$ where $f(1)=-2$ and $f(5)=-10$. Does a value $1 exist such that $f(c)=-9$?

There does not exist a value. Yes, by the Intermediate Value Theorem Yes, by the Mean Value Theorem There does not necessarily exist such a value