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\begin{document}
\begin{center}

\Huge{Limit Laws}\\
\end{center}

\noindent Let $f$ and $g$ be continuous functions, let $c$ be a constant, and $n$ be a positive integer, then if each expression exists:

\noindent $\displaystyle{\lim_{x\rightarrow a} f(x) \pm g(x)= \lim_{x\rightarrow a}f(x)\pm \lim_{x\rightarrow a} g(x)}$ \hspace{1in}  $\displaystyle{\lim_{x\rightarrow a} cf(x)= c(\lim_{x\rightarrow a}f(x))}$

\noindent $\displaystyle{\lim_{x\rightarrow a} [f(x)\cdot g(x)]= \lim_{x\rightarrow a}f(x)\cdot \lim_{x\rightarrow a} g(x)}$ \hspace{1.2in} $\displaystyle{\lim_{x\rightarrow a} \frac{f(x)}{g(x)}= \frac{\lim_{x\rightarrow a}f(x)}{ \lim_{x\rightarrow a} g(x)}}$ 

\noindent $\displaystyle{\lim_{x\rightarrow a} [f(x)^n]= [\lim_{x\rightarrow a}f(x)]^n}$ \hspace{2.0in} $\displaystyle{\lim_{x\rightarrow a} \sqrt[n]{f(x)}= \sqrt[n]{\lim_{x\rightarrow a}f(x)}}$ 

\vspace{.2in}
\normalsize
\noindent
While working in a group make sure you:
\begin{packed_itemize}
\item Expect to make mistakes but be sure to reflect/learn from them!
\item Are civil and are aware of your impact on others.
\item Assume and engage with the strongest argument while assuming best intent.
\end{packed_itemize}

\begin{enumerate}
\item The temperature (in F) of coffee is approximated by $T(x)=42+118e^{-0.04x}$.  Find $\displaystyle{\lim_{x\rightarrow 5^-} T(x)}$ and interpret it's meaning.
\vfill
\item Find $\displaystyle{\lim_{x\rightarrow 7^-} \frac{3x}{\sqrt{x+2}}}$
\vfill  
\item Find $\displaystyle{\lim_{x\rightarrow 0^+}\frac{x}{\sqrt{x+4}-2}}$
\vfill

%\item Find $\displaystyle{\lim_{x\rightarrow 0}\frac{\sin(4x)}{x}}$ 

\newpage 

\item Let $f(x)=3x^2$.  Either follow the steps below or compute the limit of the difference quotient of $f$ at $2$ directly, that is find: $$\lim_{h\rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}.$$

\begin{enumerate}
\item Find $f(2)$.
\vskip .3in
\item Find $f(2+\Delta x)$, and simplify.
\vskip .5in
\item Use (a) and (b) to find $\displaystyle{\frac{f(2+\Delta x)-f(2)}{\Delta x}}$.
\vskip 1in
\item Find $\displaystyle{\lim_{h\rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}}.$
\vskip 1in
\end{enumerate}


\item Let $g(x)=x^2-2x$.  Either follow the steps below or compute the limit of the difference quotient of $g$ at $3$ directly, that is find: $$\lim_{h\rightarrow 0} \frac{g(3+h)-g(3)}{h}.$$

\begin{enumerate}
\item Find $g(3+h)$, and simplify.
\vskip .75in
\item Use (a) to find $\displaystyle{\frac{g(3+h)-g(3)}{h}}$.
\vskip 1in
\item Find $\displaystyle{\lim_{h\rightarrow 0} \frac{g(3+h)-g(3)}{h}}.$

\end{enumerate}








\end{enumerate}
\end{document}
