1.tex

\documentclass{article}
\usepackage{multicol}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage{geometry}[top=0.5in]
\title{Basic Calculus \\
Activity II}
\author{Emu}

\begin{document}
\maketitle

\begin{enumerate}
\item Complete the following table.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$c$ & $\lim_{x \rightarrow c} 2016$ & $\lim_{x \rightarrow c} x$ \\
\hline
$-2$ & $2016$  & $-2$ \\
$-1/2$& $2016$ & $-1/2$ \\
$0$ & $2016$ & $0$\\
$3.1415$ & $2016$ & $3.1415$ \\
$10$ & $2016$ & $10$\\
$\sqrt{5}$ & $2016$ & $\sqrt{5}$ \\
\hline
\end{tabular}
\end{center}
\item Assume the following:
\[\lim_{x\rightarrow c}f(x) = \frac{3}{4},  \lim_{x\rightarrow c}g(x) = 12, \lim_{x\rightarrow c}h(x) = -3.\]
Compute the following limits (convention: $f^{-1}(x)$ is to denote the inverse of $f$, and $f(x)^{-1}$ as $1/f(x)$):
\begin{multicols}{2}
\begin{enumerate}[a.]
\item $\lim_{x\rightarrow c}(-4f(x)) = -3$
\item $\lim_{x\rightarrow c}\sqrt{12f(x)} = \pm 3$
\item $\lim_{x\rightarrow c}(g(x)-h(x)) = 15$
\item $\lim_{x\rightarrow c}(f(x)g(x)) = 9$
\item $\lim_{x\rightarrow c}(f(x)^{-1}(g(x)+h(x))) = 12$
\item $\lim_{x\rightarrow c}(h(x)^{-1}f(x)g(x)) = -3$
\item $\lim_{x\rightarrow c}(4f(x)+h(x)) = 0$
\item $\lim_{x\rightarrow c}(8f(x)-g(x)-2h(x)) = 0$
\item $\lim_{x\rightarrow c}(f(x)g(x)h(x)) = -27$
\item $\lim_{x\rightarrow c}\sqrt{-g(x)h(x)} = \pm 6$
\item $\lim_{x\rightarrow c}(h(x)^{-2}g(x)) = 4/3$
\item $\lim_{x\rightarrow c}(h(x)^{-2}g(x)f(x)) = 1$
\end{enumerate}
\end{multicols}
\item Determine whether the statement is True or False. If it is false, explain what makes it false, or provide a counter example.
\begin{enumerate}[a.]
\item If $\lim_{x\rightarrow c}f(x)$ and $\lim_{x\rightarrow c}g(x)$ both exist, then $\lim_{x\rightarrow c}(f(x)\pm g(x))$ always exist $\forall x$. Answer: True.
\item If $\lim_{x\rightarrow c}f(x)$ and $\lim_{x\rightarrow c}g(x)$ both exist, then $\lim_{x\rightarrow c}(f(x)g(x))$ always exist $\forall x$. Answer: True.
\item If $\lim_{x\rightarrow c}f(x)$ and $\lim_{x\rightarrow c}g(x)$ both exist, then $\lim_{x\rightarrow c}(f(x)/g(x))$ always exist $\forall x$. Answer:
False. Consider $f(x)=1$ and $g(x)=x$. The limit does not exist at $x=0$.
\item If $\lim_{x\rightarrow c}f(x)$ exist and  $p \in \mathbb{Z}$, then $\lim_{x\rightarrow c}f(x)^p$ always exist $\forall x$. Answer:
False. For example: $f(x)^p$ is undefined for, $f(x)=0$ and $p\in\mathbb{Z}_{\leq 0}$
\item If $\lim_{x\rightarrow c}f(x)$ exist and $n\in\mathbb{N}$, then $\lim_{x\rightarrow c}\sqrt[n]{f(x)}$ always exist $\forall x$. Answer: False. For example: $\sqrt[n]{f}$ is undefined at $(-\infty,0]$, for $n\in \{2x|x\in\mathbb{N}\}$ (this does not exclude other possibilities like $n=-1/2$).
\end{enumerate}
\item Assume the following:
\[\lim_{x\rightarrow c}f(x) = 1,  \lim_{x\rightarrow c}g(x) = -1, \lim_{x\rightarrow c}h(x) = 2.\]
Compute the following limits:
\begin{multicols}{2}
\begin{enumerate}[a.]
\item $\lim_{x\rightarrow c}(f(x)+g(x)) = 0$
\item $\lim_{x\rightarrow c}(f(x)-g(x)-h(x)) = 0$
\item $\lim_{x\rightarrow c}(3g(x)+5h(x)) = 7$
\item $\lim_{x\rightarrow c}\sqrt{f(x)} = \pm 1$
\item $\lim_{x\rightarrow c}\sqrt{g(x)} = \textrm{undefined}$
\item $\lim_{x\rightarrow c}\sqrt[3]{g(x)} = -1$
\item $\lim_{x\rightarrow c}h(x)^5 = 32$
\item $\lim_{x\rightarrow c}(h(x)^{-1}(g(x)-f(x))) = -1$
\item $\lim_{x\rightarrow c}(f(x)g(x)h(x)^2) = -4$
\item $\lim_{x\rightarrow c}(1/f(x)) = 1$
\item $\lim_{x\rightarrow c}(1/g(x)) = -1$
\item $\lim_{x\rightarrow c}(1/h(x)) = 1/2$
\item $\lim_{x\rightarrow c}(f(x)-h(x))^{-1} = -1$
\item $\lim_{x\rightarrow c}(f(x)+g(x))^{-1} = \textrm{undefined}$
\end{enumerate}
\end{multicols}
\item Assume $f(x)=x$. Evaluate:
\begin{enumerate}[a.]
\begin{multicols}{2}
\item $\lim_{x\rightarrow 4}f(x) = 4$
\item $\lim_{x\rightarrow 4}(1/f(x)) = 1/4$
\item $\lim_{x\rightarrow 4}(1/f(x)^2) = 16$
\item $\lim_{x\rightarrow 4}-\sqrt{f(x)} = -2 $
\item $\lim_{x\rightarrow 4}\sqrt{9f(x)} = \pm 6$
\item $\lim_{x\rightarrow 4}(f(x)^2-f(x)) = 12$
\item $\lim_{x\rightarrow 4}(f(x)^3+f(x)^2+2f(x)) = 88$
\item $\lim_{x\rightarrow 4}\sqrt[n]{3f(x)^2+4f(x)} = \textrm{undefined $n$}$
\item $\lim_{x\rightarrow 4} \mbox {\Large $\frac{f(x)^2-f(x)}{5f(x)}$} = 15$
\item $\lim_{x\rightarrow 4} \mbox {\Large $\frac{f(x)^2-4f(x)}{f(x)^2+4f(x)}$} = 0$
\end{multicols}
\end{enumerate}
\end{enumerate}

\end{document}