\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\usepackage{latexsym,amsmath,amssymb,epsfig}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MATH 135 -- Calculus 1}
\centerline{More on the Limit Laws}
\centerline{September 26, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Last week we introduced the Limit Laws that show how to compute limits for 
various combinations of functions when we know each ``piece'' has a limit
separately.  Today we will do a few more examples using those.
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[(1)] Assuming $\lim_{x\to 6} f(x) = 4$, compute the following limits, 
saying which of the Limit Laws you are using at every step.
\begin{enumerate}
\item[(a)]  $\lim_{x\to 6} (f(x))^2$.
\item[(b)]  $\lim_{x\to 6} \frac{1}{f(x)}$
\item[(c)]  $\lim_{x\to 6} x \sqrt{f(x)}$
\end{enumerate}
\item[(2)]  Assume that $L(a) = \lim_{x\to 0} \frac{a^x - 1}{x}$ exists for all real $a > 0$
and that $\lim_{x\to 0} a^x = 1$ for all real $a > 0$.
\begin{enumerate}
\item[(a)]  Investigate $L(2)$ and $L(3)$ numerically.  What are your guesses for 
those values?
\item[(b)]  Explain why $(ab)^x - 1 = a^x(b^x - 1) + (a^x - 1)$.
\item[(c)]  Use the result in part (b) and the Limit Laws to show $L(ab) = L(a) + L(b)$.
\item[(d)]  Using your answers from part (a), show that the formula from part (c) is reasonable
numerically when $a = 2$ and $b = 6$.  Note this means you will also need to investigate
$L(6)$ numerically. 
\item[(e)] What does the equation $L(ab) = L(a) + L(b)$ say about $L(a)$ as a function of $a$?
Do you know other functions with that property?
\end{enumerate}
\end{enumerate}
\end{document}
