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We introduce limits.

### The basic idea

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

Use the graph of $f(x) = \frac {\sin (x)}{x}$ 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 $f(x) = \frac {\sin (x)}{x}$ above to finish the following statement: “A good guess is that…”
$\lim _{x\to 0}\frac {\sin (x)}{x} = 1$. $\lim _{x\to 1}\frac {\sin (x)}{x} = 0$. $\lim _{x\to 1}f(x) = \frac {\sin (1)}{1}$. $\lim _{x\to 0}f(x) = \frac {\sin (0)}{0}=\infty$.
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 rounding to three decimal places: We may rush and say that, based on the table above, But, recall the definition of the limit: $L$ is the limit if the value of $f(x)$ is as close as one wishes to $L$ for all $x$ sufficiently close, but not equal to, $a$.

From this table we can see that $f(x)(=0)$ is as close as one wishes to $L(=0)$ for some values $x$ that are sufficiently close to $a(=0)$. But this does does not satisfy the definition of the limit, at least, not yet.

But, wait! Fill in another table. What do these two tables tell us about

$\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = 0$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right )=1$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.866$ $\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.433$ The limit does not exist.

### One-sided limits

While we have seen that $\lim _{x\to 2}\lfloor x\rfloor$ does not exist, its graph looks much “nicer” near $a=2$ than does the previous graph near $a=0$. More can be said about the function $\lfloor x\rfloor$ and its behavior near $a=2$.

### 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)
$\lim _{x\to -3^-} f(x) \begin {prompt}=\answer {6}\end {prompt}$
(d)
$\lim _{x\to 0} f(x) \begin {prompt}=\answer {DNE}\end {prompt}$
(e)
$\lim _{x\to 0^-} f(x) \begin {prompt}=\answer {-2}\end {prompt}$
(f)
$\lim _{x\to 0^+} f(x) \begin {prompt}=\answer {-1}\end {prompt}$
(g)
$f(0) \begin {prompt}=\answer {-3/2}\end {prompt}$
(h)
$f(-2) \begin {prompt}=\answer {8}\end {prompt}$
(i)
$\lim _{x\to -2^+} f(x) \begin {prompt}=\answer {2}\end {prompt}$
(j)
$\lim _{x\to -2^-} f(x) \begin {prompt}=\answer {6}\end {prompt}$
(k)
$\lim _{x\to 2^-} f(x) \begin {prompt}=\answer {7}\end {prompt}$
(l)
$\lim _{x\to 1} f(x) \begin {prompt}=\answer {3}\end {prompt}$