\DocumentMetadata{
  lang          = en,
  pdfversion    = 2.0,
  pdfstandard   = UA-2,
  pdfstandard   = A-4f,
  tagging-setup = {math/setup=mathml-SE, extra-modules={verbatim-mo}}
}
% This is in AMSLaTeX.
% On NeXT and darkwing, use LaTeX.
\documentclass[11pt]{amsart}
% \renewcommand{\baselinestretch}{1.27}
% \usepackage{showkeys}
% \usepackage{pdfsync}
% \usepackage{showlabels}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage[all]{xy}
\usepackage{graphicx}
\usepackage{unicode-math}

\hyphenation{Hirsh-berg}

\numberwithin{equation}{section}

\def\today{\number\day\space\ifcase\month\or   January\or February\or
   March\or April\or May\or June\or   July\or August\or September\or
   October\or November\or December\fi\   \number\year}

\theoremstyle{definition}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{cnj}[thm]{Conjecture}
\newtheorem{cnv}[thm]{Convention}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{ntn}[thm]{Notation}
\newtheorem{exa}[thm]{Example}
\newtheorem{pbm}[thm]{Problem}
\newtheorem{wrn}[thm]{Warning}
\newtheorem{qst}[thm]{Question}
\newtheorem{exr}[thm]{Exercise}
% \newenvironment{pff}{{\emph{Proof:}}}{\QED}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqr}{\begin{eqnarray*}}
\newcommand{\eeqr}{\end{eqnarray*}}
\newcommand{\bal}{\begin{align*}}
\newcommand{\eal}{\end{align*}}
\newcommand{\bei}{\begin{itemize}}
\newcommand{\eei}{\end{itemize}}
\newcommand{\limi}[1]{\lim_{{#1} \to \infty}}

\newcommand{\af}{\alpha}
\newcommand{\bt}{\beta}
\newcommand{\gm}{\gamma}
\newcommand{\dt}{\delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\zt}{\zeta}
\newcommand{\et}{\eta}
\newcommand{\ch}{\chi}
\newcommand{\io}{\iota}
\newcommand{\te}{\theta}
\newcommand{\ld}{\lambda}
\newcommand{\sm}{\sigma}
\newcommand{\kp}{\kappa}
\newcommand{\ph}{\varphi}
\newcommand{\ps}{\psi}
\newcommand{\rh}{\rho}
\newcommand{\om}{\omega}
\newcommand{\ta}{\tau}

\newcommand{\Gm}{\Gamma}
\newcommand{\Dt}{\Delta}
\newcommand{\Et}{\Eta}
\newcommand{\Th}{\Theta}
\newcommand{\Ld}{\Lambda}
\newcommand{\Sm}{\Sigma}
\newcommand{\Ph}{\Phi}
\newcommand{\Ps}{\Psi}
\newcommand{\Om}{\Omega}

\newcommand{\Q}{{\mathbb{Q}}}
\newcommand{\Z}{{\mathbb{Z}}}
\newcommand{\R}{{\mathbb{R}}}
\newcommand{\C}{{\mathbb{C}}}
% \newcommand{\N}{{\mathbb{N}}}
\newcommand{\N}{{\mathbb{Z}}_{> 0}}
\newcommand{\Nz}{{\mathbb{Z}}_{\geq 0}}

\pagenumbering{arabic}

\renewcommand{\Re}{{\operatorname{Re}}}
\renewcommand{\Im}{{\operatorname{Im}}}

\newcommand{\inv}{{\operatorname{inv}}}
\newcommand{\tsr}{{\operatorname{tsr}}}
\newcommand{\RR}{{\operatorname{RR}}}
\newcommand{\Tr}{{\operatorname{Tr}}}
\newcommand{\id}{{\operatorname{id}}}
\newcommand{\ev}{{\operatorname{ev}}}
\newcommand{\sint}{{\operatorname{int}}}
\newcommand{\diam}{{\operatorname{diam}}}
\newcommand{\dist}{{\operatorname{dist}}}
\newcommand{\sa}{{\operatorname{sa}}}
\newcommand{\spec}{{\operatorname{sp}}}
\newcommand{\Prim}{{\operatorname{Prim}}}
\newcommand{\diag}{{\operatorname{diag}}}
\newcommand{\supp}{{\operatorname{supp}}}
\newcommand{\rank}{{\operatorname{rank}}}
\newcommand{\sgn}{{\operatorname{sgn}}}
\newcommand{\spn}{{\operatorname{span}}}
\newcommand{\card}{{\operatorname{card}}}
\newcommand{\Aut}{{\operatorname{Aut}}}
\newcommand{\Ad}{{\operatorname{Ad}}}
\newcommand{\Aff}{{\operatorname{Aff}}}
\newcommand{\rc}{{\operatorname{rc}}}
\newcommand{\mdim}{{\operatorname{mdim}}}

\newcommand{\cA}{{\mathcal{A}}}
\newcommand{\cB}{{\mathcal{B}}}
\newcommand{\cC}{{\mathcal{C}}}
\newcommand{\cD}{{\mathcal{D}}}
\newcommand{\cE}{{\mathcal{E}}}
\newcommand{\cF}{{\mathcal{F}}}
\newcommand{\cG}{{\mathcal{G}}}
\newcommand{\cH}{{\mathcal{H}}}
\newcommand{\cI}{{\mathcal{I}}}
\newcommand{\cJ}{{\mathcal{J}}}
\newcommand{\cK}{{\mathcal{K}}}
\newcommand{\cL}{{\mathcal{L}}}
\newcommand{\cM}{{\mathcal{M}}}
\newcommand{\cN}{{\mathcal{N}}}
\newcommand{\cO}{{\mathcal{O}}}
\newcommand{\cP}{{\mathcal{P}}}
\newcommand{\cQ}{{\mathcal{Q}}}
\newcommand{\cR}{{\mathcal{R}}}
\newcommand{\cS}{{\mathcal{S}}}
\newcommand{\cT}{{\mathcal{T}}}
\newcommand{\cU}{{\mathcal{U}}}
\newcommand{\cV}{{\mathcal{V}}}
\newcommand{\cW}{{\mathcal{W}}}
\newcommand{\cX}{{\mathcal{X}}}
\newcommand{\cY}{{\mathcal{Y}}}
\newcommand{\cZ}{{\mathcal{Z}}}

\newcommand{\dirlim}{\varinjlim}
% \newcommand{\dirlim}{\displaystyle \lim_{\longrightarrow}}
\newcommand{\invlim}{\varprojlim}
\newcommand{\Mi}{M_{\infty}}

\newcommand{\andeqn}{\qquad {\mbox{and}} \qquad}
\newcommand{\QED}{\rule{0.4em}{2ex}}
\newcommand{\ts}[1]{{\textstyle{#1}}}
\newcommand{\ds}[1]{{\displaystyle{#1}}}
\newcommand{\ssum}[1]{{\ts{ {\ds{\sum}}_{#1} }}}
\newcommand{\Sq}[2]{(#1_{#2})_{#2 \in \N}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\wolog}{without loss of generality}
\newcommand{\Wolog}{Without loss of generality}
\newcommand{\Tfae}{The following are equivalent}
\newcommand{\tfae}{the following are equivalent}
\newcommand{\ifo}{if and only if}
\newcommand{\wrt}{with respect to}

\newcommand{\ca}{C*-algebra}
\newcommand{\uca}{unital C*-algebra}
\newcommand{\usa}{unital subalgebra}
\newcommand{\hm}{homomorphism}
\newcommand{\uhm}{unital homomorphism}
\newcommand{\am}{automorphism}
\newcommand{\im}{isomorphism}
\newcommand{\fd}{finite dimensional}
\newcommand{\tst}{tracial state}
\newcommand{\hsa}{hereditary subalgebra}

\newcommand{\pj}{projection}
\newcommand{\mops}{mutually orthogonal \pj s}
\newcommand{\nzp}{nonzero projection}
\newcommand{\mvnt}{Murray-von Neumann equivalent}
\newcommand{\mvnc}{Murray-von Neumann equivalence}

\newcommand{\ct}{continuous}
\newcommand{\cfn}{continuous function}
\newcommand{\nbhd}{neighborhood}
\newcommand{\cms}{compact metric space}
\newcommand{\cpt}{compact Hausdorff}
\newcommand{\chs}{compact Hausdorff space}
\newcommand{\hme}{homeomorphism}
\newcommand{\mh}{minimal homeomorphism}
\newcommand{\mf}{manifold}
\newcommand{\cmf}{compact smooth \mf}

\newcommand{\cp}{crossed product}
\newcommand{\tgca}{transformation group \ca}
\newcommand{\tggp}{transformation group groupoid}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\rsz}[1]{\raisebox{0ex}[0.8ex][0.8ex]{$#1$}}
\newcommand{\subrsz}[1]{\raisebox{0ex}[0.8ex][0.8ex]{${\scriptstyle{#1}}$}}

% Temporary abbreviations:

\renewcommand{\S}{\subset}
\newcommand{\ov}{\overline}
\newcommand{\SM}{\setminus}
\newcommand{\I}{\infty}
\newcommand{\E}{\varnothing}

\newcommand{\Lem}[1]{Lemma~\ref{#1}}
\newcommand{\Def}[1]{Definition~\ref{#1}}
\newcommand{\Thm}[1]{Theorem~\ref{#1}}
\newcommand{\Prp}[1]{Proposition~\ref{#1}}
\newcommand{\Exa}[1]{Example~\ref{#1}}

\title{Math 251: summary of topics}

% \author{N.~Christopher Phillips}

\date{29 March 2026}


\begin{document}

\maketitle

% \setcounter{section}{-1}

\indent
% Introduction goes here.

\section{Summary of the main topics}\label{Sec_5315_Outline}

See the separate page about arithmetic you are expected to be able to
do without a calculator.
Also, remember that notation counts!
See the separate notation page.
% {\tt{(Need to add more here.)}} % MORE

\bigskip

Here is a list of most of the topics of the course
(not necessarily complete).
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_Meaning_L}
Understanding the meaning of the limit,
both one sided and two sided, at finite values and at $\pm \I$,
and which have finite values, are $\pm \I$,
or do not exist and are not even $\pm \I$.
Understanding includes recognizing them from graphs
and numerical interpretation, for example,
how the values of $f (2{.}1)$, $f (2{.}01)$, $f (2{.}001)$, etc.\  %
and $f (1{.}9)$, $f (1{.}99)$, $f (1{.}999)$ etc.\  %
relate to $\ds{ \lim_{x \to 2} f (x) }$.
%
\item\label{I_5315_Calc_L}
Limit laws, and finding limits of all the kinds above.
We have seen many methods for doing this.
See Section~\ref{Sec_5315_Lim} for more.
%
\item\label{I_5315_Ctg}
Continuity, in particular, its relation to limits
and recognizing continuity from the graph.
%
\item\label{I_5315_Meaning}
Definition, meaning, and interpretation of the derivative:
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_Dfn}
$\ds{f' (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h} }$,
if the limit exists.
%
\item\label{I_5315_Sec}
Limit of the slopes of secant lines to the graph.
%
\item\label{I_5315_Tan}
Slope of the tangent line to the graph.
%
\item\label{I_5315_D_Gr}
Recognizing differentiability (existence of a tangent line)
from the graph.
%
\item\label{I_5315_RofC}
Instantaneous rate of change.
The most common example is velocity, but there are many other
rates of change, some of which don't involve time.
(Acceleration is included here: it is the rate of change of
velocity.)
%
\item\label{I_5315_LinApp}
Linear approximation.
%
% \item\label{I_5315_???}
% QQQ
%
\end{enumerate}
%
\item\label{I_5315_Appl}
Applications of derivatives.
Some examples, not already in~(\ref{I_5315_Meaning}):
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_Graph}
Shape of a graph:
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_CrtP}
Critical points.
%
\item\label{I_5315_LocMM}
Local minimums and maximums.
%
\item\label{I_5315_IncrDecr}
Relation between the sign of the derivative
and whether the function is increasing or decreasing.
%
\item\label{I_5315_Concavity}
Relations between the second derivative and concavity.
%
\item\label{I_5315_InflP}
Inflection points.
%
\item\label{I_5315_2Test}
Second derivative test for a local minimum or maximum.
%
\end{enumerate}
%
\item\label{I_5315_MinMx}
Finding the maximum or minimum
of a function on a closed bounded interval.
%
\item\label{I_5315_MMWord}
Applied maximization and minimization problems (word problems).
Gary Larson cartoon on word problems:

\vspace{1ex}
\centerline{
\includegraphics[width=11cm]{Hells_Library288dpi}
}
\vspace{1ex}
%
\item\label{I_5315_RelR}
Related rates (also word problems).
%
\item\label{I_5315_MVT}
Using the Mean Value Theorem to get information about the change in
a function from information about its derivative.
\end{enumerate}
%
\end{enumerate}
%


\indent

\section{Additional comments on limits}\label{Sec_5315_Lim}

\indent
{\textbf{Warning:}} using L'Hopital's Rule when its hypotheses don't apply
will result in getting the entire problem wrong.
This could cost {\textbf{half or more of a letter grade}} on the final exam.

{\textbf{Warning:}} you will be expected to know that $\sin (0) = 0$,
$\cos (0) = 1$, $\ln (e) = 1$, $a^0 = 1$ when $a \neq 0$,
etc.\  {\textbf{without using a calculator.}}
Getting these wrong may result in a misidentification
of an indeterminate form, and thus getting an entire limit problem wrong,
costing {\textbf{half or more of a letter grade}} on the final exam.

\bigskip

We have seen at least seven methods for calculating limits
at finite values of the variable.
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_Sub}
Direct substitution:
if $f$ is \ct{} at~$a$, then $\ds{ \lim_{x \to a} f (x) = f (a)}$.
{\textbf{Always}} check first if this case applies!
%
\item\label{I_5315_Cancel}
Cancellation of common factors in a fraction.
%
\item\label{I_5315_Rat}
Rationalization in a ratio containing roots.
%
\item\label{I_5315_VertA}
Identifying vertical asymptotes, for example in a limit of the
form $\ds{\lim_{x \to a} \frac{f (x)}{g (x)} }$ in which $g (a) = 0$
but $f (a) \neq 0$.
(Vertical asymptotes also occur elsewhere.
For example, $\ds{\lim_{x \to 0^{+}} \ln (x) = - \I}$.)
%
\item\label{I_5315_LH}
L'Hopital's Rule, but {\textbf{only}} for indeterminate forms!
%
\item\label{I_5315_Sq}
Squeeze Theorem.
%
\item\label{I_5315_Osc}
The limit does not exist because the function oscillates too much.
For example, $\ds{\lim_{x \to 0} \sin \left( \frac{1}{x} \right) }$
does not exist.
(But $\ds{\lim_{x \to 0} x \sin \left( \frac{1}{x} \right) } = 0$
by the Squeeze Theorem.)
%
\end{enumerate}
%

As an example, specifically for fractions,
consider $\ds{\lim_{x \to a} \frac{f (x)}{g (x)} }$.
For the purposes of illustration,
assume that $f$ and $g$ are both continuous at~$a$.
Then there are three possibilities.
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_DNonZ}
$g (a) \neq 0$.
Then the function $\ds{q (x) = \frac{f (x)}{g (x)}}$ is \ct{} at~$a$,
so
\[
\lim_{x \to a} \frac{f (x)}{g (x)}
 = \lim_{x \to a} q (x)
 = q (a)
 = \frac{f (a)}{g (a)}.
\]
This is true even if $f (a) = 0$.
%
\item\label{I_5315_DZ}
$g (a) = 0$ but $f (a) \neq 0$.
Then the function $\ds{q (x) = \frac{f (x)}{g (x)}}$ has a vertical
asymptote at $x = a$.
(The limit $\ds{\lim_{x \to \pi / 2} \tan (x)}$ is of this type.)
This is {\textbf{not}} an indeterminate form, since no finite limit
is possible.
You expect the one sided limits
$\ds{\lim_{x \to a^{+}} \frac{f (x)}{g (x)} }$
and $\ds{\lim_{x \to a^{-}} \frac{f (x)}{g (x)} }$ to be $\I$ or $- \I$.
If they aren't both the same, then
$\ds{\lim_{x \to a} \frac{f (x)}{g (x)} }$ does not exist, even
as $\I$ or~$- \I$.
%
\item\label{I_5315_Indet}
$g (a) = 0$ and $f (a) = 0$.
This is an indeterminate form, and methods like cancelling common
factors, rationalization, or L'Hopital's Rule might be useful.
You can certainly try L'Hopital's Rule, but {\textbf{only}} in this case..
%
\end{enumerate}
It may be that instead $\ds{\lim_{x \to a} f (x) = \pm \I}$,
$\ds{\lim_{x \to a} g (x) = \pm \I}$, or both.
Among these cases,
only $\ds{\lim_{x \to a} f (x) = \pm \I}$ {\textbf{and}}
$\ds{\lim_{x \to a} g (x) = \pm \I}$ is an indeterminate form,
namely ``$\dfrac{\pm \I}{\pm \I}$''.
But some other combinations, for example
$\ds{\lim_{x \to a} f (x) = \pm \I}$
and $\ds{\lim_{x \to a} g (x) = 0}$,
that is, the form ``$\dfrac{\pm \I}{0}$'', give vertical asymptotes.


We have seen a similar collection of methods for calculating limits
at $\pm \I$.
First, here are examples of limits at infinity which we
implicitly already knew before this course,
for example just by looking at the graphs:
\[
\lim_{x \to \I} x
  = \lim_{x \to \I} x^2
  = \lim_{x \to \I} x^3
  = \lim_{x \to \I} \sqrt{x} = \I,
  \qquad {\mbox{etc.}}
\]
\[
\lim_{x \to \I} x
  = \lim_{x \to \I} x^3
  = \lim_{x \to \I} x^5 = - \I,
  \qquad {\mbox{etc.}}
\]
\[
\lim_{x \to \I} x^2
  = \lim_{x \to \I} x^4
  = \lim_{x \to \I} x^6 = \I,
  \qquad {\mbox{etc.}}
\]
\[
\lim_{x \to \I} e^x = \I,
\quad
\lim_{x \to - \I} e^x = 0,
\quad
\lim_{x \to \I} \ln (x) = \I,
\quad
\lim_{x \to \I} \arctan (x) = \frac{\pi}{2},
  \quad {\mbox{etc.}}
\]
\[
\lim_{x \to \I} \frac{1}{x}
 = \lim_{x \to \I} \frac{1}{x^2}
 = \lim_{x \to \I} \frac{1}{x^3}
 = \lim_{x \to \I} \frac{1}{\sqrt{x}} = 0,
  \qquad {\mbox{etc.}}
\]
\[
\lim_{x \to - \I} \frac{1}{x}
 = \lim_{x \to - \I} \frac{1}{x^2}
 = \lim_{x \to - \I} \frac{1}{x^3} = 0,
  \qquad {\mbox{etc.}}
\]
%

Here are some of the methods we have seen.
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_5315_Sub_I}
Limits of combinations of the functions appearing above
gotten by using the limit laws and
reasoning with $\pm \I$ in ways analogous to the limit laws.
(These are the analogs of both direct substitution in continuous
functions and some cases of identifying vertical asymptotes.)
%
\item\label{I_5315_Canc_I}
Multiplying both numerator and denominator of a fraction
by $\dfrac{1}{x^n}$, or similar things (for example, $e^{- 2 x}$).
This is the analog of cancellation of common factors in a fraction.
%
\item\label{I_5315_LH_I}
L'Hopital's Rule, but {\textbf{only}} for indeterminate forms!
%
\item\label{I_5315_Sq_I}
Squeeze Theorem.
%
\item\label{I_5315_Osc_I}
The limit does not exist because the function oscillates too much.
For example, $\ds{\lim_{x \to \I} \sin (x) }$
does not exist.
(But $\ds{\lim_{x \to 0} \frac{\sin (x)}{x} } = 0$
by the Squeeze Theorem.)
%
\end{enumerate}
%

% \section{???}\label{Sec_5315_???}

% \indent

\end{document}
