# opencalculus/open-calculus

Updated to v1.02

Began work on Chapter 2 (Limits)
 @@ -10,10 +10,11 @@ \usepackage{amsmath} \usepackage{anysize} \usepackage{hyperref} \usepackage{graphicx} \usepackage{bigstrut} \usepackage{framed} \setlength{\parindent}{0cm} \marginsize{0.5in}{0.5in}{0.5in}{0.5in} \everymath{\displaystyle} \linespread{1.1} @@ -37,7 +38,7 @@ Dixon \textsc{Crews}\\[0.5cm] \small \texttt{\href{mailto:dixon@opencalculus.org}{dixon@opencalculus.org}} \\ [3.5cm] {\large Version 1.01}\\[1cm] {\large Version 1.02}\\[1cm] {\large March 2014}\\[3cm] \vfill @@ -63,6 +64,48 @@ \subsection{What is Calculus?} \section{Limits} \subsection{Introduction} \textbf{Definition:} The limit of $f(x)$ as $x$ approaches $a$ equals $L$ \begin{align*} \lim_{x \to a} f(x) = L \end{align*} if we can make the values of $f(x)$ as close to $L$ as we like by taking $x$ to be close to $a$ but not equal to $a$. It is important to note that we do not consider the case where $x = a$ when evaluating limits. $f(x)$ may not be defined at $x = a$, so we are just concerned with the behavior of $f(x)$ as it approaches $x = a$. \\ \subsubsection{Numerical Limit Evaluation} We can numerically evaluate limits by creating a table of values and letting the independent variable (usually $x$ or $t$) get very close to our $a$ value. Let's look at an example. \\ \textbf{Evaluate} $\lim_{x \to 2} \sqrt{x}$ given \hspace{1em} \begin{tabular}{c | c} $x$ & $\sqrt{x}$ \\ \hline 1.98 & 1.4071 \\ 1.99 & 1.4107 \\ 2.00 & ? \\ 2.01 & 1.4177 \\ 2.02 & 1.4213 \\ \end{tabular} Let's choose 1.414 as our limit. $\sqrt{2}$ to four decimal places is actually 1.4142, so this is quite a good approximation! \subsection{One-Sided Limits} We can also look at limits from both directions. If we're evaluating a limit and see that we approach a different value from the left than we do when approaching from the right, the limit \textbf{does not exist (DNE)}. \\ A left-sided limit is denoted by changing $a$ to $a^-$: $\lim_{x \to a^-} f(x) = L$ \\ Similarly, a right-sided limit is denoted by changing $a$ to $a^+$: $\lim_{x \to a^+} f(x) = L$ \\ \begin{center} \begin{framed} $\lim_{x \to a} f(x) = L$ \textbf{iff} $\lim_{x \to a^-} f(x) = L$ \textbf{and} $\lim_{x \to a^+} f(x) = L$ \end{framed} \end{center} \subsection{Properties of Limits} \subsection{Continuity} \subsection{Infinite Limits} \section{Differentiation} \section{Integration}