Objective 2 - Vertical Asymptotes - Ximera

$\newcommand{\arraycolsep}[3]{} \newcommand{\dd}[2][]{\frac{d #1}{d #2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\veci}{\vec{\hat{\bf\imath}}} \newcommand{\vecj}{\vec{\hat{\bf\jmath}}} \newcommand{\utan}{\vec{\hat{t}}} \newcommand{\unormal}{\vec{\hat{n}}} \newenvironment{amatrix}[1]{% \begin{bmatrix} }{% \end{bmatrix} }$

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Link to section in online textbook.

Introduction video describing holes/vertical asymptotes without limits.

When we learned about the domain of rational functions, we set the denominator equal to 0 and solved. This gave us all values that the function is not defined for. As we saw in Objective 1, some of these $x$ -values are holes of the function. The rest are vertical asymptotes of the function, or vertical lines where the function approaches positive or negative infinity.

$\graph [rectangular]{f(x) = 1/((x-1)(x-3))}$

Unlike holes, vertical asymptotes affect the function around where they are defined. With left- and right-sided limits, we can determine how the function is behaving near these vertical lines.

Vertical Asymptotes of a Rational Function

A rational function has a vertical asymptote $x = a$ (vertical line) if

$\lim _{x \rightarrow a^{-}} f(x) = \pm \infty \text { or } \lim _{x \rightarrow a^{+}} f(x) = \pm \infty$

Thus, to determine if a rational function has any vertical asymptotes, we need to factor the denominator and evaluate the limit. If a one-sided limit is positive or negative infinity, it is a vertical asymptote.

Practice with the questions below.

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function1num}}{\sage {function1denom}}$

Vertical Asymptote: the vertical line $x = \answer [format=string]{NA}$ .

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function2num}}{\sage {function2denom}}$

Vertical Asymptote: the vertical line $x = \answer [tolerance=0.05]{\sage {answer2}}$ .

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function3num}}{\sage {function3denom}}$

Vertical Asymptote: the vertical line $x = \answer [tolerance=0.05]{\sage {answer3a}}$ and $x = \answer [tolerance=0.05]{\sage {answer3b}}$ .

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function4num}}{\sage {function4denom}}$

Vertical Asymptote: the vertical line $x = \answer [tolerance=0.05]{\sage {answer4a}}$ , $x = \answer [tolerance=0.05]{\sage {answer4b}}$ , and $x = \answer [tolerance=0.05]{\sage {answer4c}}$ .

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function5num}}{\sage {function5denom}}$

Vertical Asymptote: the vertical line $x = \answer [tolerance=0.05]{\sage {answer5b}}$ and $x = \answer [tolerance=0.05]{\sage {answer5c}}$ .

Find all vertical asymptotes of the rational function below. If they do not exist, answer “NA”.

$f(x) = \frac {\sage {function6num}}{\sage {function6denom}}$

Vertical Asymptote: the vertical line $x = \answer [format=string]{NA}$ .

Main takeaway: Values not in the domain of the function can be one of two things:

Vertical Asymptotes: Values the function will come close to, but will not touch at. These are vertical lines $x = a$ , where $a$ makes the denominator zero. The limit of the function at these points are $\pm \infty$ .
Holes: Values that only affect the function exactly at that point (rather than nearby by vertical asymptotes). The limit of the function at these points are $\frac {0}{0}$ .