Browse files

Updated to v1.02

Began work on Chapter 2 (Limits)
  • Loading branch information...
dixoncrews committed Mar 2, 2014
1 parent 2fd2edb commit a5423b5402aa019b221ec7b5c66b5648d1f95c3e
Showing with 47 additions and 4 deletions.
  1. +1 −1 .gitignore
  2. BIN OpenCalculus.pdf
  3. +46 −3 OpenCalculus.tex
@@ -30,4 +30,4 @@
Binary file not shown.
@@ -10,10 +10,11 @@
@@ -37,7 +38,7 @@
Dixon \textsc{Crews}\\[0.5cm]
\small \texttt{\href{}{}} \\ [3.5cm]
{\large Version 1.01}\\[1cm]
{\large Version 1.02}\\[1cm]
{\large March 2014}\\[3cm]
@@ -63,6 +64,48 @@ \subsection{What is Calculus?}
\textbf{Definition:} The limit of $f(x)$ as $x$ approaches $a$ equals $L$
\lim_{x \to a} f(x) = L
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 \\
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 $a^+$: $\lim_{x \to a^+} f(x) = L$ \\
$\lim_{x \to a} f(x) = L$ \textbf{iff} $\lim_{x \to a^-} f(x) = L$ \textbf{and} $\lim_{x \to a^+} f(x) = L$
\subsection{Properties of Limits}
\subsection{Infinite Limits}

0 comments on commit a5423b5

Please sign in to comment.