## Section3.3Computing Limits: Graphically

In this section we look at an example to illustrate the concept of a limit graphically.

The graph of a function $f(x)$ is shown below. We analyze the behaviour of $f(x)$ around $x=-5\text{,}$ $x=-2\text{,}$ $x=-1$ and $x=0\text{,}$ and $x=4\text{.}$

Observe that $f(x)$ is indeed a function (it passes the Vertical Line Test). We now analyze the function at each point separately.

x=-5: Observe that at $x=-5$ there is no closed circle, thus $f(-5)$ is undefined. From the graph we see that as $x$ gets closer and closer to $-5$ from the left, then $f(x)$ approaches $2\text{,}$ so

\begin{equation*} \lim_{x\to -5^-}f(x)=2\text{.} \end{equation*}

Similarly, as $x$ gets closer and closer $-5$ from the right, then $f(x)$ approaches $-3\text{,}$ so

\begin{equation*} \lim_{x\to -5^+}f(x)=-3\text{.} \end{equation*}

As the right-hand limit and left-hand limit are not equal at $-5\text{,}$ we know that

\begin{equation*} \lim_{x\to -5}f(x)\mbox{ does not exist.} \end{equation*}

x=-2: Observe that at $x=-2$ there is a closed circle at $0\text{,}$ thus $f(-2)=0\text{.}$ From the graph we see that as $x$ gets closer and closer to $-2$ from the left, then $f(x)$ approaches $3.5\text{,}$ so

\begin{equation*} \lim_{x\to -2^-}f(x)=3.5\text{.} \end{equation*}

Similarly, as $x$ gets closer and closer $-2$ from the right, then $f(x)$ again approaches $3.5\text{,}$ so

\begin{equation*} \lim_{x\to -2^+}f(x)=3.5\text{.} \end{equation*}

As the right-hand limit and left-hand limit are both equal to $3.5\text{,}$ we know that

\begin{equation*} \lim_{x\to -2}f(x)=3.5\text{.} \end{equation*}

Do not be concerned that the limit does not equal 0. This is a discontinuity, which is completely valid, and will be discussed in a later section.

We leave it to the reader to analyze the behaviour of $f(x)$ for $x$ close to $-1$ and $0\text{.}$

Summarizing, we have:

\begin{equation*} \begin{array}{cccc} f(-5) \mbox{ is undefined} \amp f(-2)=0 \amp f(-1)=-2 \amp f(0)=-2 \\ \hline \ds{\lim_{x\to -5^-}f(x)=2} \amp \ds{\lim_{x\to -2^-}f(x)=3.5} \amp \ds{\lim_{x\to -1^-}f(x)=0} \amp \ds{\lim_{x\to 0^-}f(x)=-2} \\ ~\amp ~\amp ~\amp ~\\ \ds{\lim_{x\to -5^+}f(x)=-3} \amp \ds{\lim_{x\to -2^+}f(x)=3.5} \amp \ds{\lim_{x\to -1^+}f(x)=-2} \amp \ds{\lim_{x\to 0^+}f(x)=-2}\\ ~\amp ~\amp ~\amp ~\\ \ds{\lim_{x\to -5}f(x)=\mbox{DNE} } \amp \ds{\lim_{x\to -2}f(x)=3.5} \amp \ds{\lim_{x\to -1}f(x)=\mbox{DNE} } \amp \ds{\lim_{x\to 0}f(x)=-2} \end{array} \end{equation*}
##### Exercises for Section 3.3.

Evaluate the expressions by reference to this graph:

1. $\ds \lim_{x\to 4} f(x)$

$\lim\limits_{x \to 4} f(x) = 8$
2. $\ds \lim_{x\to -3} f(x)$

$\lim\limits_{x \to -3} f(x) = 6$
3. $\ds \lim_{x\to 0} f(x)$

$\lim\limits_{x \to 0} f(x)$ DNE, since the left and right side limits are not the same.
4. $\ds \lim_{x\to 0^-} f(x)$

$\lim\limits_{x \to 0^{-}} f(x) = -2\text{.}$ Note that this is not equal to $f(0)\text{,}$ but is instead the value which $f$ is approaching as $x \to 0$ from the left.
5. $\ds \lim_{x\to 0^+} f(x)$

$\lim\limits_{x \to 0^{+}} f(x) = -1$
6. $\ds f(-2)$

$f(-2) = 8$
7. $\ds \lim_{x\to 2^-} f(x)$

$\lim\limits_{x \to 2^{-}} f(x) = 7$
8. $\ds \lim_{x\to -2^-} f(x)$

$\lim\limits_{x \to -2^{-}} f(x) = 6$
9. $\ds \lim_{x\to 0} f(x+1)$

$\lim\limits_{x \to 0} f(x+1) = \lim\limits_{x \to 1} f(x) = 3$
10. $\ds f(0)$

$f(0) = -1.5$
11. $\ds \lim_{x\to 1^-} f(x-4)$

$\lim\limits_{x \to 1^{-}} f(x-4) = \lim\limits_{x \to -3^{-}} f(x) = 6$
12. $\ds \lim_{x\to 0^+} f(x-2)$

$\lim\limits_{x \to 0^{+}} f(x-2) = \lim\limits_{x \to -2^{+}} f(x) = 2$

Evaluate the expressions by reference to this graph:

1. $\lim\limits_{x\to -1} f(x)$

$\lim\limits_{x\to -1} f(x) =$ DNE, since the left and right-side limits are not equal.
2. $\lim\limits_{x\to 0^+} f(x)$

$\lim\limits_{x\to 0^+} f(x) = 2$
3. $\lim\limits_{x\to 1} f(x)$
$\lim\limits_{x\to 0^+} f(x) = 2$
4. $f(1)$
$f(1) = 0$
5. $\lim\limits_{x\to 2} f(x)$
$\lim\limits_{x\to 2} f(x) = 2$