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We introduce limits.

### The basic idea

Consider the function graphed below. While $f(x)$ is undefined at $x=0$, we can still plot $f(x)$ at other values near $x = 0$.

Use the graph of $f$ above to answer the following question: What is $f(0)$?
$0$ $f(0)$ $1$ $f(0)$ is undefined it is impossible to say

Nevertheless, we can see that as $x$ approaches zero, $f(x)$ approaches one. From this setting we come to our definition of a limit.

Use the graph of $f$ above to finish the following statement: “A good guess is that …”
$\lim _{x\to 0}f(x) = 1$. $\lim _{x\to 1}f(x) = 0$. $\lim _{x\to 1}f(x) = f(1)$. $\lim _{x\to 0}f(x) = f(0)$.
Consider the following graph of $y=f(x)$ Use the graph to evaluate the following. Write DNE if the value does not exist.
(a)
$f(-2) \begin {prompt}=\answer {1}\end {prompt}$
(b)
$\lim _{x\to -2}f(x)\begin {prompt}=\answer {1}\end {prompt}$
(c)
$f(-1) \begin {prompt}=\answer {2}\end {prompt}$
(d)
$\lim _{x\to -1}f(x) \begin {prompt}=\answer {2}\end {prompt}$
(e)
$f(0) \begin {prompt}=\answer {-2}\end {prompt}$
(f)
$\lim _{x\to 0} f(x) \begin {prompt}=\answer {0}\end {prompt}$
(g)
$f(1) \begin {prompt}=\answer {DNE}\end {prompt}$
(h)
$\lim _{x\to 1} f(x) \begin {prompt}=\answer {-2}\end {prompt}$

### Limits might not exist

Limits might not exist. Let’s see how this happens.

Tables can be used to help guess limits, but one must be careful.

Consider $f(x) = \sin \left (\frac {\pi }{x}\right )$. Fill in the tables below (to three decimal places): What do the tables tell us about
$\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = 0$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right )=1$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.866$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.433$ it is unclear what the tables are telling us about the limit

### One-sided limits

While we have seen that $\lim _{x\to 2}\lfloor x\rfloor$ does not exist, more can still be said.

### When you put this all together

One-sided limits help us talk about limits.

Evaluate the expressions by referencing the graph below. Write DNE if the limit does not exist.
(a)
$\lim _{x\to 4} f(x) \begin {prompt}=\answer {8}\end {prompt}$
(b)
$\lim _{x\to -3} f(x)\begin {prompt}=\answer {6}\end {prompt}$
(c)
$\lim _{x\to 0} f(x) \begin {prompt}=\answer {DNE}\end {prompt}$
(d)
$\lim _{x\to 0^-} f(x) \begin {prompt}=\answer {-2}\end {prompt}$
(e)
$\lim _{x\to 0^+} f(x) \begin {prompt}=\answer {-1}\end {prompt}$
(f)
$f(-2) \begin {prompt}=\answer {8}\end {prompt}$
(g)
$\lim _{x\to 2^-} f(x) \begin {prompt}=\answer {7}\end {prompt}$
(h)
$\lim _{x\to -2^-} f(x) \begin {prompt}=\answer {6}\end {prompt}$
(i)
$\lim _{x\to 0} f(x+1) \begin {prompt}=\answer {3}\end {prompt}$
(j)
$f(0) \begin {prompt}=\answer {-3/2}\end {prompt}$
(k)
$\lim _{x\to 1^-} f(x-4) \begin {prompt}=\answer {6}\end {prompt}$
(l)
$\lim _{x\to 0^+} f(x-2) \begin {prompt}=\answer {2}\end {prompt}$