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\markright{Chapter 2: Worksheet 3}
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\Bf{Chapter 2: Worksheet 3 \hfill ANSWERS \hfill MA 131}

\begin{enumerate}
\item
	\begin{enumerate}
	\item $u = x^3, h(x) = \sin u$ or $g(x) = x^3, f(x) = \sin x$
	\item $u = \sin x, h = u^3$
	\item $u = \sin x,  h =3^u$
	\item $u = x^3 + 3, h = \sin u$
	\item $u = (x^4 + x^2 + 1), h = u^7$
	\end{enumerate}

\item 
	\begin{enumerate}
	\item $u = x + 1, v = \ln u, h = \tan v$
	\item $u = x^2, v = \sin u, h = 2^v$
	\item $u = \sin x, v = \sin u, h = \sin v$
	\end{enumerate}

\item 
	\begin{enumerate}
	\item 0
	\item 0  (see Figure 1 and 2).  Note $\sin \pi/2 = 1.0$ and $\sin (\sin (\sin \pi /2 )) = 0.745624$.
	Since $\pi/2$ produces the largest value, once we have 
	the result there, it isn't hard to realize that the function goes to 0 at \Em{every} $x$ as we
	take more and more compositions.
	\end{enumerate}

\begin{figure}[htb]
\epsfysize 80pt
\centerline{\epsffile{ans3p3a.eps}}
\caption{4 compositions of $\sin$ with itself}
\end{figure}

\begin{figure}[htb]
\epsfysize 80pt
\centerline{\epsffile{ans3p3b.eps}}
\caption{128 compositions of $\sin$ with itself}
\end{figure}

\item   Merely evaluating the function to get the limit is justified when the function is \Em{continuous}.  If you don't see this important point, study the definition of continuity carefully and/or ask your instructor.
	\begin{enumerate}
	\item Rational functions are continuous except at points where the denominator vanishes.  Hence we may just ``plug in'' for this example: $\lim_{x \rightarrow 5}$ {\Large $\frac{x + 5}{x^4 + x^2 + 1} = \frac{5 + 5}{625 + 25 + 1} = \frac{10}{651}$}
	\item Here we have the favorite case in calculus where numerator and denominator both vanish at $a$.  Because of the denominator, the function is \Em{not} continuous at $x = 5$.  However, in evaluating the limit, we do not have to consider this point, so we may employ factoring for $x \neq 5$ to get $\lim_{x \rightarrow 5} (x + 5) = 10$.
	\item No problem, just ``plug in'': $\lim_{x \rightarrow 5} \sqrt{x^6 - 5} = \sqrt{5^5 - 5} = \sqrt{5 \cdot 624}$
	\end{enumerate}

\item
	\begin{enumerate}
	\item 1 (For example, make a Table or use the Program of the Week)
	\item 0
	\item $e \approx 2.71828$
	\end{enumerate}

\end{enumerate}
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