Learning Outcomes
- Determine limits of functions approaching vertical asymptotes.
Section 1.6 Limits with infinite outputs (LT6)
Subsection 1.6.1 Activities
Activity 1.6.1.
Consider the graph in Figure 33.
(a)
Which of the following best describes the limit as \(x\) approaches zero in the graph?
- The limit is 0
- The limit is positive infinity
- The limit does not exist
- This limit is negative infinity
(b)
Which of the following best describes the relationship between the line \(x=0 \) and the graph of the function?
- The line \(x=0\) is a horizontal asymptote for the function
- The function is not continuous at the point \(x=0\)
- The function is moving away from the line \(x=0\)
- The function is getting closer and closer to the line \(x=0\)
- The function has a jump in outputs around \(x=0\)
Definition 1.6.2.
A function has a vertical asymptote at \(x=a\) when
\begin{equation*}
\lim_{x\to a} f(x) = + \infty
\end{equation*}
or
\begin{equation*}
\lim_{x\to a} f(x) = - \infty
\end{equation*}
The limit being equal to positive infinity means that we can make the output of \(f(x)\) as large a positive number as we want as long as we are sufficiently close to \(x=a\text{.}\) Similarly, the limit being equal to negative infinity means that we can make the output of \(f(x)\) as large a negative number as we want as long as we are sufficiently close to \(x=a\text{.}\)
Activity 1.6.3.
Select all of the following graphs which illustrate functions with vertical asymptotes.
Remark 1.6.4.
If \(x=a\) is a vertical asymptote for the function \(f(x) \text{,}\) the function \(f(x)\) is not defined at \(x=a\text{.}\) As \(f(a)\) does not exist, the function is NOT continuous at \(x=a\text{.}\) Moreover, the function's output tends to plus or minus infinity and so the limit is not equal to a number.
Activity 1.6.5.
Notice that as \(x\) goes to 0, the value of \(x^2\) goes to 0 but the value of \(1/x^2\) goes to infinity. What is the best explanation for this behavior?
- When dividing by an increasingly small number we get an increasing big number
- When dividing by an increasingly large number we get an increasing small number
- A rational function always has a vertical asymptote
- A rational function always has a horizontal asymptote
Remark 1.6.6.
Informally, we say that the limit of "\(\frac{1}{0}\)" is infinite. Notice that this could be either positive or negative infinity, depending on how whether the outputs are becoming more and more positive or more and more negative as we approach zero.
Activity 1.6.7.
Consider the rational function \(f(x) = \frac{2}{x-3} \text{.}\) Which of the following options best describes the limits as x approaches \(3\) from the right and from the left?
- As \(x \to 3^+\text{,}\) the limit DNE, but as \(x \to 3^-\) the limit is \(-\infty\text{.}\)
- As \(x \to 3^+\text{,}\) the limit is \(+\infty\text{,}\) but as \(x \to 3^-\) the limit is \(-\infty\text{.}\)
- As \(x \to 3^+\text{,}\) the limit is \(+\infty\text{,}\) but as \(x \to 3^-\) the limit is \(+\infty\text{.}\)
- As \(x \to 3^+\text{,}\) the limit is \(-\infty\text{,}\) but as \(x \to 3^-\) the limit is \(-\infty\text{.}\)
- As \(x \to 3^+\text{,}\) the limit DNE and as \(x \to 3^-\) the limit DNE.
Remark 1.6.8.
When considering a ratio of functions \(f(x)/g(x) \text{,}\) the inputs \(a\) where \(g(a)=0\) are not in the domain of the ratio. If \(g(a)=0 \) but \(f(a) \) is not equal to 0, then \(x=a\) is a vertical asymptote.
Activity 1.6.9.
Consider the function \(f(x)=\frac{x^2-1}{x-1}\text{.}\) The line \(x=1\) is NOT a vertical asymptote for \(f(x)\text{.}\) Why?
- When \(x\) is not equal to \(1\text{,}\) we can simplify the fraction to \(x-1\text{,}\) so the limit is \(1\text{.}\)
- When \(x\) is not equal to \(1\text{,}\) we can simplify the fraction to \(x+1\text{,}\) so the limit is \(2\text{.}\)
- The function is always equal to \(x+1\text{.}\)
- The function is always equal to \(x-1\text{.}\)
Remark 1.6.10.
Recall the definition of a hole from Definition 1.3.16. In Activity 1.6.9 we have a hole at \(x=1\text{.}\)
Activity 1.6.11.
Find all the vertical asymptotes of the following rational functions.
(a)
\(y = \frac{3x-4}{7x+1}\)
(b)
\(y= \frac{x^2+10x+24}{x^2-2x+1}\)
(c)
\(y= \frac{(x^2-4)(x^2+1)}{x^6}\)
(d)
\(y= \frac{2x+1}{2x^2+8x-10}\)
Activity 1.6.12.
Explain and demonstrate how to find the value of each limit.
(a)
\begin{equation*}
\lim_{x\to-3^- } \frac{{\left(x + 4\right)}^{2} {\left(x - 2\right)}}{{\left(x + 3\right)} {\left(x - 5\right)}}
\end{equation*}
(b)
\begin{equation*}
\lim_{x\to-3^+ } \frac{{\left(x + 4\right)}^{2} {\left(x - 2\right)}}{{\left(x + 3\right)} {\left(x - 5\right)}}
\end{equation*}
(c)
\begin{equation*}
\lim_{x\to-3 } \frac{{\left(x + 4\right)}^{2} {\left(x - 2\right)}}{{\left(x + 3\right)} {\left(x - 5\right)}}
\end{equation*}
Activity 1.6.13.
The graph below represents the function \(f(x) = \displaystyle\frac{(x+2)(x+4)}{x^2+3x-4}\text{.}\)
(a)
Explain the behavior of \(f(x)\) at \(x=-4 \text{.}\)
(b)
Find the vertical asymptote(s) of \(f(x)\text{.}\) First, guess it from the graph. Then, prove that your guess is right using algebra.
(c)
Find the horizontal asymptote(s) of \(f(x)\text{.}\) First, guess it from the graph. Then, prove that your guess is right using algebra.
(d)
Use limit notation to describe the behavior of \(f(x)\) at its asymptotes.
Activity 1.6.14.
Consider the following rational function.
\begin{equation*}
r(x) = \frac{ 5 \, {\left(x - 3\right)} {\left(x - 6\right)}^{3} }{ 6 \, {\left(x + 2\right)}^{3} {\left(x - 3\right)} }
\end{equation*}
(a)
Explain how to find the horizontal asymptote(s) of \(r(x)\text{,}\) if there are any. Then express your findings using limit notation.
(b)
Explain how to find the hole(s) of \(r(x)\text{,}\) if there are any. Then express your findings using limit notation.
(c)
Explain how to find the vertical asymptote(s) of \(r(x)\text{,}\) if there are any. Then express your findings using limit notation.
(d)
Draw a rough sketch of \(r(x)\) that showcases all the limits that you have found above.
Activity 1.6.15.
You want to draw a function with all these properties.
- \(\displaystyle \displaystyle \lim_{x\to 3} f(x)=5\)
- \(\displaystyle f(3)=0 \)
- \(\displaystyle \displaystyle \lim_{x\to 0^{-}} f(x) = -\infty\)
- \(\displaystyle \displaystyle \lim_{x\to 0^{+}} f(x) = 0\)
- \(\displaystyle \displaystyle \lim_{x\to +\infty} f(x) = 2\)
(a)
At which \(x\) values will the limit not exist?
(b)
What are the asymptotes of this function?
(c)
At which \(x\) values will the function be discontinuous?
(d)
Draw the graph of one function with all the properties above. Make sure that your graph is a function! You only need to draw a graph, writing a formula would be very challenging!
