Limits

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty} \newcommand {\todo }[0]{} \newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{}$

language of behavior

Let

\[ F(t) = \begin{cases} \frac {(t-2)(t+3)}{(t-2)} &\text {if $t \ne 2$,}\\ 1 &\text {if $t = 2$}. \end{cases} \]

Graph of $y = F(t)$.

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

From the surrounding values, we expected $F(2) = 5$. Instead, we got $F(2) = 1$.

There is nothing wrong with $F$. It is what it is. Our head just saw a pattern and then one of the points didn’t follow our pattern.

Our graph describes all of this.

Now, we would just like to describe all of this with algebra (our written language).

Function notation allows us to describe what is there, $F(2) = 1$.

Now, we need a way to communicate our expectation. We expected $F(2) = 5$.

Limits will be our language for expectations.

The story of limits is a long story. There are many surprising twists and turns to limits. We’ll begin the story in Precalculus. However, we’ll leave the nuts and bolts to Calculus. Calculus will get into the nitty gritty details.

For us, here, it is a communication issue.

In the example above we want to say that the behavior of $F$ “around” $2$ leads us to expects $F(2)=5$.

Our language for this will look like

\[ \lim \limits _{t \to 2} F(t) = 5\]

This is pronounced as:

“The limit of $F$ as $t$ approaches $2$ equals $5$”

Limits are written like this:

\[ \lim \limits _{t \to 2} F(t) = 5\]

It starts out with “lim” as an abbreviation for limit.
underneath “lim” you say where the domain is headed. You give the variable, then an arrow, then the domain number. It all fits underneath the “lim”. Next to all of that, you put the mathematical expression you are investigating. This is often just the function name, but it will become other things.
Then an equals sign.
Then the expected value.

This is called a two-sided limit, because the behavior on either side of $2$ conveys an expectation of $5$.

We have also seen that our expectations may differ on either side.

Graph of $z = G(r)$.

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

We now have two expectations for the same domain number. The left side of our brain is expecting $G(2)=5$ and that expectation is satisfied. The right side of our brain is expecting $G(2)=1$ and that expectation comes up empty.

From the left, the negative direction, we expect $G(2)=5$
From the right, the positive direction, we expect $G(2)=1$

To communicate these one-sided expectations, we include “-” and “+” as superscripts to our limit notation

\[ \lim \limits _{t \to 2^-} G(t) = 5 \]

\[ \lim \limits _{t \to 2^+} G(t) = 1 \]

Just communicating function behavior and expectations. We are just developing language and notation and symbols to communicate a full story of the functions were are analyzing.

Don’t fret. Calculus will continue this story to any depth of minutia that you desire. Calculus will extend this story to any number of dimensions that interest you.

ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Broken Values

2026-01-22 21:13:48