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\noindent {\bf Question}

\noindent For each of the functions $f(x)$ given below, consider
the sequence constructed by setting $x_{n+1}=f(x_n)$ for $n\ge 0$
and taking $x_0=c$.  Determine whether $\{ x_n\}$ converges or
diverges, and note that this may depend on the initial choice of
$c$. Where possible, calculate the limit when it exists.
\begin{enumerate}
\item $f(x)=x+3$;
\item $f(x)=\frac{1}{3} x + \frac{3}{4}$;
\item $f(x)=\frac{2}{5} x + \frac{1}{5}$;
\item $f(x)=10-x$;
\item $f(x)=\sqrt{3x}$;
\item $f(x)=\frac{1}{2} \left( x+\frac{c}{x} \right)$;
\item $f(x)= \frac{1}{2} (x+4)$;
\end{enumerate}

\medskip

\noindent {\bf Answer}


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