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\noindent {\bf Question}

\noindent Explain {\bf exactly} what is meant by the following
statements:
\begin{enumerate}
\item $\lim_{n\rightarrow\infty} 3^{2n-1} =\infty$;
\item $\lim_{n\rightarrow\infty} (1-2n) =-\infty$;
\item $\lim_{n\rightarrow\infty} e^{-n} =0$;
\end{enumerate}

\medskip

\noindent {\bf Answer}

\noindent (This is an exercise in writing out the definition of
the convergence or divergence of a sequence for a triple of
specific examples.  Note that we are not asked to determine
whether the given statements are true or false, or to prove them
if they are true, but just to write them down.)
\begin{itemize}
\item for every $\varepsilon >0$, there exists $M$ so that $3^{2n-1}
>\varepsilon$ for all $n >M$.
\item for every $\varepsilon >0$, there exists $M$ so that $1-2n
<-\varepsilon$ for all $n >M$.
\item for every $\varepsilon >0$, there exists $M$ so that $| e^{-n}
-0| <\varepsilon$ for all $n >M$.
\end{itemize}

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