Objective 1 - Limit Notation

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Interpret the notation for limits.

Link to section in online textbook.

Intro video for limit notation

Our College Algebra textbook gives a light introduction to “arrow notation” when talking about limits. This is a great starting point to understand what exactly a limit is.

Symbol	Meaning

$x \rightarrow a^{-}$	$x$ approaches $a$ from the left
$x \rightarrow a^{+}$	$x$ approaches $a$ from the right
$x \rightarrow \infty$	$x$ approaches infinity
$x \rightarrow -\infty$	$x$ approaches negative infinity

This notation works for the output of a function as well! So if we say $f(x) \rightarrow \infty$ , we mean that the output of the function approaches infinity. We’ve already seen this with end behavior of polynomials. For example, if we wanted to describe the end behavior of $f(x) = x^2$ , we would say “ $f(x) \rightarrow \infty$ as $x \rightarrow \infty$ and as $x \rightarrow -\infty$ . The limit notation condenses this phrase.

$\lim _{x \rightarrow a} (f(x)) = L$ means “as $x \rightarrow a$ , $f(x) \rightarrow L$ ”.

Let’s practice. Use this Desmos link to answer the following questions about $f(x)=\frac {1}{x}$ .

As $x$ approaches infinity, what happens to the $y$ value of $\frac {1}{x}$ ?

$\lim _{x \rightarrow \infty }\left (\frac {1}{x}\right ) = \answer {0}$

As $x$ approaches negative infinity, what happens to the $y$ value of $\frac {1}{x}$ ?

$\lim _{x \rightarrow -\infty }\left (\frac {1}{x}\right ) = \answer {0}$

Looking at the graph, you are probably wondering what we would say about the limit as $x$ approaches $0$ of $f(x) = \frac {1}{x}$ . We will deal with that in the next objective. For the rest of this objective, we’ll practice interpreting the limit notation.

Translate the phrase “ $\frac {x+3}{x^2-9}$ approaches $-\frac {1}{6}$ as $x$ approaches $-3$ ” into limit notation.

$\lim _{\answer {x} \rightarrow \answer {-3}}\left ( \answer {\sage {(x+3)/(x**2-9)}} \right ) = \answer {-\frac {1}{6}}$

Translate the phrase “as $x$ approaches infinity, $-(x+2)^3(x-3)^2$ approaches negative infinity” into limit notation.

$\lim _{\answer {x} \rightarrow \answer {\sage {Infinity} } }\left ( \answer {\sage {-(x+2)**3 * (x-3)**2}} \right ) = \answer {\sage {-Infinity}}$

Translate the phrase “as $x$ approaches $3$ , $\frac {1}{(x-3)^2}$ approaches infinity” into limit notation.

$\lim _{\answer {x} \rightarrow \answer {3}}\left ( \answer {\sage {1/(x-3)**2}} \right ) = \answer {\sage {Infinity}}$