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\begin_layout Title
Asymptotes
\end_layout

\begin_layout Subtitle
\noindent
In this module we give a new definition to an idea you have seen before
 by defining 
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 asymptotes
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 in terms of limits.
\end_layout

\begin_layout BeginFrame
Defining the problem
\end_layout

\begin_layout Standard
\noindent
Do the following functions have 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

? 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec y=1/x$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec y=\ln x$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec y=(3x^{2}+x+1)/(2x^{2}-2)$
\end_inset

 
\end_layout

\begin_layout Standard
If so, what kind of asymptote, and where? If not, why not?
\end_layout

\begin_layout BeginFrame
Horizontal asymptotes
\end_layout

\begin_layout Standard
In the module on limits at infinity, we noticed that some functions have
 a finite value for 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\pm\infty}f(x)}.
\]

\end_inset

 One example was 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\pm\infty}\frac{1}{x}=0}.
\]

\end_inset

 
\begin_inset Float figure
placement h!
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\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
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{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/plot_1_over_x_horiz_as.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
Horizontal asymptotes
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{definition}[
\end_layout

\end_inset

(Horizontal asymptote)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
If 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}f(x)=L$
\end_inset

 where 
\begin_inset Formula $\ec L$
\end_inset

 is a real number, or if 
\begin_inset Formula $\ec\lim_{x\rightarrow-\infty}f(x)=L$
\end_inset

 where 
\begin_inset Formula $\ec L$
\end_inset

 is a real number, then the line 
\begin_inset Formula $\ec y=L$
\end_inset

 is a 
\series bold

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 horizontal asymptote
\series default
 of 
\begin_inset Formula $\ec f(x)$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{definition}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
In the previous example, 
\begin_inset Formula $\ec y=0$
\end_inset

 is a horizontal asymptote of 
\begin_inset Formula $\ec1/x$
\end_inset

.
\end_layout

\begin_layout BeginFrame
\noindent
Horizontal asymptotes: 
\begin_inset Formula $\ln x$
\end_inset


\end_layout

\begin_layout Standard
\noindent
On the other hand, what is 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}\ln x$
\end_inset

? 
\begin_inset Float figure
placement h!
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sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/plot_lnx_plain.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Here 
\begin_inset Formula $\ec L=1$
\end_inset

 and 
\begin_inset Formula $\ec\varepsilon=1$
\end_inset

; 
\begin_inset Formula $\ec x>e^{2}$
\end_inset

 implies that 
\begin_inset Formula $\ec\ln x>L+\varepsilon$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\color white
Let 
\begin_inset Formula $\varepsilon=1$
\end_inset

.
 No matter what value you choose for 
\begin_inset Formula $L$
\end_inset

, eventually 
\begin_inset Formula $|\ln x-L|>\varepsilon$
\end_inset

 for 
\emph on
every
\emph default
 value of 
\begin_inset Formula $x$
\end_inset

.
\end_layout

\begin_layout BeginFrame
Horizontal asymptotes: 
\begin_inset Formula $\ln x$
\end_inset


\end_layout

\begin_layout Standard
\noindent
On the other hand, what is 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}\ln x$
\end_inset

? 
\begin_inset Float figure
placement h!
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sideways false
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\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
centering{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/lnx_and_epsilon.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Here 
\begin_inset Formula $\ec L=1$
\end_inset

 and 
\begin_inset Formula $\ec\varepsilon=1$
\end_inset

; 
\begin_inset Formula $\ec x>e^{2}$
\end_inset

 implies that 
\begin_inset Formula $\ec\ln x>L+\varepsilon$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\ec\varepsilon=1$
\end_inset

.
 No matter what value you choose for 
\begin_inset Formula $\ec L$
\end_inset

, eventually 
\begin_inset Formula $\ec|\ln x-L|>\varepsilon$
\end_inset

 for 
\emph on
every
\emph default
 value of 
\begin_inset Formula $\ec x$
\end_inset

.
\end_layout

\begin_layout BeginFrame
Horizontal asymptotes: 
\begin_inset Formula $\ln x$
\end_inset


\end_layout

\begin_layout Standard
\noindent
On the other hand, what is 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}\ln x$
\end_inset

? Let 
\begin_inset Formula $\ec\varepsilon=1$
\end_inset

.
 No matter what value you choose for 
\begin_inset Formula $\ec L$
\end_inset

, eventually 
\begin_inset Formula $\ec|\ln x-L|>\varepsilon$
\end_inset

 for 
\emph on
every
\emph default
 value of 
\begin_inset Formula $\ec x$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/lnx_and_epsilon.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

\end_layout

\end_inset

 Here 
\begin_inset Formula $\ec L=1$
\end_inset

 and 
\begin_inset Formula $\ec\varepsilon=1$
\end_inset

; 
\begin_inset Formula $\ec x>e^{2}$
\end_inset

 implies that 
\begin_inset Formula $\ec\ln x>L+\varepsilon$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Since 
\begin_inset Formula $\ec\ln x$
\end_inset

 is a one-to-one function, if 
\begin_inset Formula $\ec x>\exp(L+\varepsilon)$
\end_inset

 then 
\begin_inset Formula $\ec|\ln x-L|>\varepsilon$
\end_inset

.
\end_layout

\begin_layout BeginFrame
Horizontal asymptotes: third example
\end_layout

\begin_layout Standard
\noindent
The third example is
\begin_inset Formula 
\[
\ec{f(x)=\frac{3x^{2}+x+1}{2x^{2}-1}}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
To find the limit, we rewrite this expression, multiplying numerator and
 denominator by 
\begin_inset Formula $1/x^{2}$
\end_inset

 to obtain
\begin_inset Formula 
\[
\ec f(x)=\frac{3x^{2}+x+1}{2x^{2}-1}\cdot\frac{\frac{1}{x^{2}}}{\frac{1}{x^{2}}}=\frac{3+\frac{1}{x}+\frac{1}{x^{2}}}{2-\frac{1}{x^{2}}}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
This allows us to say that
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow0}\frac{3+\cancelto{0}{\frac{1}{x}}+\cancelto{0}{\frac{1}{x^{2}}}}{2-\cancelto{0}{\frac{1}{x^{2}}}}=\frac{3}{2}.
\]

\end_inset


\end_layout

\begin_layout BeginFrame
Vertical asymptotes
\end_layout

\begin_layout Standard
\noindent
Now we move to 
\begin_inset ERT
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{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 vertical asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 The graph of 
\begin_inset Formula $\ec1/x$
\end_inset

 has a vertical asymptote at 
\begin_inset Formula $\ec x=0$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/plot_1_over_x_vert_as.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
How can we describe this using limits?
\end_layout

\begin_layout BeginFrame
Vertical asymptotes
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Now we move to vertical asymptotes.
 The graph of 
\begin_inset Formula $1/x$
\end_inset

 has a vertical asymptote at 
\begin_inset Formula $x=0$
\end_inset

.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/plot_1_over_x_vert_as.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
How can we describe this using limits?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Notice that the 
\begin_inset Formula $y$
\end_inset

-values increase dramatically when the 
\begin_inset Formula $x$
\end_inset

-values approach 0.
\end_layout

\begin_layout BeginFrame
Vertical asymptotes: definition
\end_layout

\begin_layout Standard
That suggests the following definition.
 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{definition}[(Vertical asymptotes)]
\end_layout

\end_inset


\end_layout

\begin_layout Standard
A 
\series bold

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 vertical asymptote
\series default
 occurs at 
\begin_inset Formula $\ec x=a$
\end_inset

 when
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a^{+}}f(x)=\pm\infty}\quad\mathrm{or}\quad{\ec\lim_{x\rightarrow a^{-}}f(x)=\pm\infty}.
\]

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{definition}
\end_layout

\end_inset

 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_1_over_x_vert_as.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Limit 
\emph on
of
\emph default
 infinity
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Here we encounter a new idea:
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a}f(x)=\pm\infty}.
\]

\end_inset

 Intuitively it makes sense, but we'd best give a precise definition for
 it.
\end_layout

\begin_layout BeginFrame
\noindent
A limit 
\emph on
of
\emph default
 infinity?
\end_layout

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\begin_inset ERT
status open

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{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Here we encounter a new idea:
\begin_inset Formula 
\[
{\lim_{x\rightarrow a}f(x)=\pm\infty}.
\]

\end_inset

 Intuitively it makes sense, but we'd best give a precise definition for
 it.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We want to capture the notion that 
\end_layout

\begin_layout Itemize
when the 
\begin_inset Formula $x$
\end_inset

 values draw arbitrarily close to 
\begin_inset Formula $\ec a$
\end_inset

, 
\end_layout

\begin_layout Itemize
the 
\begin_inset Formula $y$
\end_inset

 values increase (or decrease) without bound.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/plot_1_over_x_vert_as.png
	lyxscale 20
	scale 30

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Go back and review the precise definitions of (a) a finite limit, and (b)
 a limit 
\emph on
at
\emph default
 infinity.
 See if you can combine those two ideas into the definition of a limit 
\emph on
of
\emph default
 infinity.
\end_layout

\begin_layout Pause
\noindent

\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{definition}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

[(Limit of )]
\end_layout

\end_inset

 
\begin_inset Formula $\lim_{x\rightarrow a}=\infty$
\end_inset

 if 
\end_layout

\begin_layout Itemize
for all 
\begin_inset Formula $\ec N\in\mathbb{R}$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset

 such that 
\end_layout

\begin_deeper
\begin_layout Itemize
for all 
\begin_inset Formula $\ec x$
\end_inset

 satisfying 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

), then 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f(x)>N$
\end_inset

 for all 
\begin_inset Formula $\ec f(x)$
\end_inset

 (except maybe 
\begin_inset Formula $\ec f(a)$
\end_inset

).
 
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{definition}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Study the definition to make sure you understand how it captures the notion.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A limit of 
\begin_inset Formula $-\infty$
\end_inset

?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We can build a similar definition for a limit of 
\begin_inset Formula $\ec-\infty$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{definition}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

[(Limit of
\end_layout

\end_inset

 
\begin_inset Formula $-\infty$
\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

)]
\end_layout

\end_inset

 
\begin_inset Formula $\lim_{x\rightarrow a}=-\infty$
\end_inset

 if 
\end_layout

\begin_layout Itemize
for all 
\begin_inset Formula $\ec N\in\mathbb{R}$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset

 such that 
\end_layout

\begin_deeper
\begin_layout Itemize
for all 
\begin_inset Formula $\ec x$
\end_inset

 satisfying 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

), then 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f(x)<N$
\end_inset

 for all 
\begin_inset Formula $\ec f(x)$
\end_inset

 (except maybe 
\begin_inset Formula $\ec f(a)$
\end_inset

).
 
\end_layout

\end_deeper
\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{definition}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
Detecting vertical asymptotes: division by zero?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{definition}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

[(Vertical asymptotes)]
\end_layout

\end_inset

 A 
\series bold

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 vertical asymptote
\series default
 occurs at 
\begin_inset Formula $\ec x=a$
\end_inset

 when
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a^{+}}f(x)=\pm\infty}\quad\mathrm{or}\quad{\ec\lim_{x\rightarrow a^{-}}f(x)=\pm\infty}.
\]

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{definition}
\end_layout

\end_inset

 In the case of 
\begin_inset Formula $\ec1/x$
\end_inset

, we can notice this from division by zero.
 If a function has the form 
\begin_inset Formula $\ec f(x)/g(x)$
\end_inset

 such that 
\begin_inset Formula $\ec x\rightarrow a$
\end_inset

 implies that 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec g(x)$
\end_inset

 approaches zero, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f(x)$
\end_inset

 approaches a nonzero constant, 
\end_layout

\begin_layout Standard
\noindent
then the function has a vertical asymptote at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 We won't prove this observation, but you should think about why it's true.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{theorem}
\end_layout

\end_inset

 If 
\begin_inset Formula $\ec f(x)\rightarrow1/0$
\end_inset

 when 
\begin_inset Formula $\ec x\rightarrow a$
\end_inset

 from the left or from the right,
\begin_inset Newline newline
\end_inset

 then 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=\infty$
\end_inset

 or 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=-\infty$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{theorem}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Detecting vertical asymptotes: division by zero?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{theorem}
\end_layout

\end_inset

 If 
\begin_inset Formula $\ec f(x)\rightarrow1/0$
\end_inset

 when 
\begin_inset Formula $\ec x\rightarrow a$
\end_inset

 from the left or from the right,
\begin_inset Newline newline
\end_inset

 then 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=\infty$
\end_inset

 or 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=-\infty$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{theorem}
\end_layout

\end_inset

 It's important that the numerator approaches a 
\emph on
nonzero
\emph default
 constant.
 If both the numerator and the denominator approach zero, then there may
 be a hole instead of an asymptote, as with the example
\begin_inset Formula 
\[
\ec\frac{3x+1}{x}
\]

\end_inset

in the module on limits at infinity.
\end_layout

\begin_layout BeginFrame
\noindent
Detecting vertical asymptotes
\end_layout

\begin_layout Standard
\noindent
Not all functions with vertical asymptotes fit this mold.
 In other cases, you must perform a careful analysis before you can decide
 whether there is a vertical asymptote.
 This is the case with 
\begin_inset Formula $\ec\ln x$
\end_inset

: graphical and numerical checks suggest that it has a vertical asymptote
 at 
\begin_inset Formula $\ec x=0$
\end_inset

.
 Since 
\begin_inset Formula $\ec\ln x$
\end_inset

 is undefined for 
\begin_inset Formula $\ec x<0$
\end_inset

, we need check only that 
\begin_inset Formula $\ec\lim_{x\rightarrow0^{+}}\ln x=-\infty$
\end_inset

.
 
\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.01 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0001 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ln x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-2.3026 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-4.6052 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-6.9078 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-9.2103 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 1
use_makebox 0
width "8cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_lnx.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
Detecting vertical asymptotes
\end_layout

\begin_layout Standard
\noindent
If we want to use the precise definition and show that 
\begin_inset Formula $\ec\ln x<N$
\end_inset

 for any real number 
\begin_inset Formula $\ec N$
\end_inset

 once we are close enough to zero, the choice 
\begin_inset Formula $\ec x=e^{-N}$
\end_inset

 suffices.
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 1
use_makebox 0
width "13cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_lnx_smallerthan-3.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

\end_layout

\end_inset

 If 
\begin_inset Formula $\ec x<e^{-1}$
\end_inset

, then 
\begin_inset Formula $\ec\ln x<-1$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Other asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Some functions have asymptotes that are neither horizontal nor vertical.
 These can be 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes, or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 nonlinear
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes.
 In each case, the function has the form 
\begin_inset Formula $\ec a(x)+z(x)$
\end_inset

 where as 
\begin_inset Formula $\ec x\rightarrow\infty$
\end_inset

, 
\begin_inset Formula $\ec z(x)\rightarrow0$
\end_inset

.
\end_layout

\begin_layout BeginFrame
Other asymptotes
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Some functions have asymptotes that are neither horizontal nor vertical.
 These can be 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes, or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 nonlinear
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes.
 In each case, the function has the form 
\begin_inset Formula $a(x)+z(x)$
\end_inset

 where as 
\begin_inset Formula $x\rightarrow\infty$
\end_inset

, 
\begin_inset Formula $z(x)\rightarrow0$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
For example, if 
\begin_inset Formula 
\[
{\ec f(x)=\frac{x^{2}+1}{x+1}},
\]

\end_inset

 polynomial division shows that
\begin_inset Formula 
\[
{\ec f(x)=x-1+\frac{2}{x+1}}.
\]

\end_inset

 Here 
\begin_inset Formula $\ec a(x)=x-1$
\end_inset

 and 
\begin_inset Formula $\ec z(x)=\frac{2}{x+1}$
\end_inset

.
 As 
\begin_inset Formula $\ec x\rightarrow\pm\infty$
\end_inset

, 
\begin_inset Formula $\ec\frac{2}{x+1}\rightarrow0$
\end_inset

 so 
\begin_inset Formula $\ec x-1$
\end_inset

 is an 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 oblique asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Other asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Some functions have asymptotes that are neither horizontal nor vertical.
 These can be 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes, or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 nonlinear
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes.
 In each case, the function has the form 
\begin_inset Formula $a(x)+z(x)$
\end_inset

 where as 
\begin_inset Formula $x\rightarrow\infty$
\end_inset

, 
\begin_inset Formula $z(x)\rightarrow0$
\end_inset

.
\end_layout

\begin_layout Standard
For example, 
\begin_inset Formula 
\[
{f(x)=\frac{x^{2}+1}{x+1}=x-1+\frac{2}{x+1}}.
\]

\end_inset

 Here 
\begin_inset Formula $a(x)=x-1$
\end_inset

 and 
\begin_inset Formula $z(x)=\frac{2}{x+1}$
\end_inset

.
 As 
\begin_inset Formula $x\rightarrow\pm\infty$
\end_inset

, 
\begin_inset Formula $\frac{2}{x+1}\rightarrow0$
\end_inset

 so 
\begin_inset Formula $x-1$
\end_inset

 is an 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

oblique asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 1
use_makebox 0
width "8cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_oblique_asymptote.png
	lyxscale 20
	scale 20

\end_inset


\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

\end_layout

\end_inset

 Oblique asymptote 
\begin_inset Formula $\ec x-1$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Other asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Some functions have asymptotes that are neither horizontal nor vertical.
 These can be 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes, or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 nonlinear
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 asymptotes.
 In each case, the function has the form 
\begin_inset Formula $a(x)+z(x)$
\end_inset

 where as 
\begin_inset Formula $x\rightarrow\infty$
\end_inset

, 
\begin_inset Formula $z(x)\rightarrow0$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
For a nonlinear example, take 
\begin_inset Formula 
\[
{\ec f(x)=\ln x+\frac{1}{x}}.
\]

\end_inset

 Here 
\begin_inset Formula $\ec a(x)=\ln x$
\end_inset

 and 
\begin_inset Formula $\ec z(x)=\frac{1}{x}$
\end_inset

.
 As 
\begin_inset Formula $\ec x\rightarrow\pm\infty$
\end_inset

, 
\begin_inset Formula $\ec\frac{1}{x}\rightarrow0$
\end_inset

 so 
\begin_inset Formula $\ec\ln x$
\end_inset

 is a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 nonlinear asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 1
use_makebox 0
width "8cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
\align center
 
\begin_inset Graphics
	filename figures/plot_nonlinear_asymptote.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Box ovalbox
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "100col%"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

\end_layout

\end_inset

 The nonlinear asymptote 
\begin_inset Formula $\ec\ln x$
\end_inset

.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Find the asymptotes of
\begin_inset Formula 
\[
{\ec\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We start by looking for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

vertical asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 The only place this can occur is when the denominator is zero (all the
 other 
\begin_inset Formula $x$
\end_inset

-values are defined); that is,
\begin_inset Formula 
\begin{align*}
\ec2x^{2}-2 & \ec=0\\
\ec x^{2} & \ec=1\\
\ec x & {\ec=\pm1}.
\end{align*}

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Now we consider 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

horizontal
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nonlinear asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\begin_inset Formula 
\begin{align*}
\ec\lim_{x\rightarrow\pm\infty}\frac{3x^{2}+x+1}{2x^{2}-2} & \ec=\lim_{x\rightarrow\pm\infty}\frac{\left(3x^{2}+x+1\right)\cdot\frac{1}{x^{2}}}{\left(2x^{2}-2\right)\cdot\frac{1}{x^{2}}}\\
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Multiply top and bottom by the hightest power of 
\begin_inset Formula $x$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Now we consider horizontal, oblique, or nonlinear asymptotes.
 
\begin_inset Formula 
\begin{align*}
\dims\lim_{x\rightarrow\pm\infty}\frac{3x^{2}+x+1}{2x^{2}-2} & \dims=\lim_{x\rightarrow\pm\infty}\frac{\left(3x^{2}+x+1\right)\cdot\frac{1}{x^{2}}}{\left(2x^{2}-2\right)\cdot\frac{1}{x^{2}}}\\
 & \ec=\lim_{x\rightarrow\pm\infty}\frac{3+\frac{1}{x}+\frac{1}{x^{2}}}{2-\frac{2}{x^{2}}}\\
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Distributed 
\begin_inset Formula $1/x^{2}$
\end_inset

.
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Now we consider horizontal, oblique, or nonlinear asymptotes.
 
\begin_inset Formula 
\begin{align*}
\dims\lim_{x\rightarrow\pm\infty}\frac{3x^{2}+x+1}{2x^{2}-2} & \dims=\lim_{x\rightarrow\pm\infty}\frac{3+\frac{1}{x}+\frac{1}{x^{2}}}{2-\frac{2}{x^{2}}}\\
 & \ec=\lim_{x\rightarrow\infty}\frac{3+\cancelto{0}{\frac{1}{x}}+\cancelto{0}{\frac{1}{x^{2}}}}{2-\cancelto{0}{\frac{2}{x^{2}}}}
\end{align*}

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Using 
\begin_inset Formula $\lim_{x\rightarrow\pm\infty}1/x=0$
\end_inset

 and limit properties\SpecialChar \ldots{}

\begin_inset ERT
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\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Now we consider horizontal, oblique, or nonlinear asymptotes.
 
\begin_inset Formula 
\begin{align*}
\dims\lim_{x\rightarrow\pm\infty}\frac{3x^{2}+x+1}{2x^{2}-2} & \dims=\lim_{x\rightarrow\infty}\frac{3+\cancelto{0}{\frac{1}{x}}+\cancelto{0}{\frac{1}{x^{2}}}}{2-\cancelto{0}{\frac{2}{x^{2}}}}\\
 & {\ec=\frac{3}{2}}.
\end{align*}

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
 It has a horizontal asymptote at 
\begin_inset Formula $y=3/2.$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
There are no oblique or non-linear asymptotes; long division of polynomials
 gives us
\begin_inset Formula 
\[
\ec{\frac{3x^{2}+x+1}{2x^{2}-2}=\frac{3}{2}+\frac{x+4}{2x^{2}-2}},
\]

\end_inset

 and as 
\begin_inset Formula $\ec x\rightarrow\pm\infty$
\end_inset

 this approaches 
\begin_inset Formula $\ec3/2$
\end_inset

, which corresponds to the horizontal asymptote.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A complete example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Find the asymptotes of
\begin_inset Formula 
\[
{\frac{3x^{2}+x+1}{2x^{2}-2}}.
\]

\end_inset


\end_layout

\begin_layout Standard
It has vertical asymptotes at 
\begin_inset Formula $x=\pm1$
\end_inset

.
 It has a horizontal asymptote at 
\begin_inset Formula $y=3/2.$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Combining this information and plotting a few points, we end up with this
 graph: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Graphics
	filename figures/complete_example.png
	lyxscale 20
	scale 20

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
(The graph crosses 
\begin_inset Formula $\ec y=-3/2$
\end_inset

 on the left, but still approaches it eventually.)
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We considered different ways to find the 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 asymptotes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 of a function 
\begin_inset Formula $\ec g(x)$
\end_inset

.
 All of these come from limits: 
\end_layout

\begin_layout Itemize
A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

horizontal asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 exists at 
\begin_inset Formula $\ec x=a$
\end_inset

 if 
\begin_inset Formula $\ec\lim_{x\rightarrow a}g(x)=\pm\infty$
\end_inset

.
 This is true even if the limit is only from one side.
 
\end_layout

\begin_layout Itemize
A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

vertical asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 exists at 
\begin_inset Formula $\ec y=L$
\end_inset

 if 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}g(x)=L$
\end_inset

.
 This is true for 
\begin_inset Formula $\ec x\rightarrow-\infty$
\end_inset

 as well.
 
\end_layout

\begin_layout Itemize
An 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

oblique
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nonlinear asymptote
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset Formula $\ec f(x)$
\end_inset

 exists if 
\begin_inset Formula $\ec g(x)=a(x)+z(x)$
\end_inset

 where 
\begin_inset Formula $\ec z\rightarrow0$
\end_inset

 when 
\begin_inset Formula $\ec x\rightarrow0$
\end_inset

.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
