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\noindent {\bf Question}

\noindent Explain {\bf exactly} what is meant by the following
statements:
\begin{enumerate}
\item $\lim_{x\rightarrow 1} (2x)^4 =16$;
\item $\lim_{x\rightarrow -3} (3x^2+e^x) =81+e^{-3}$;
\end{enumerate}
\medskip

\noindent {\bf Answer}

\noindent (Note that we are not asked to determine whether the
statement is correct or not, and if it is correct we are not asked
to prove it. This is an exercise in writing down the definition of
$\lim_{x\rightarrow a} f(x) =L$ for specific values of $a$ and $L$
and a specific function $f(x)$.)
\begin{enumerate}
\item For every $\varepsilon >0$, there exists $\delta >0$ so that if
$0 <|x -1| <\delta$, then $|(2x)^4 -16| <\varepsilon$.
\item For every $\varepsilon >0$, there exists $\delta >0$ so that if
$0 <|x - (-3)| = | x+3| <\delta$, then $|(3x^2 +e^x) - (81 +
e^{-3})| <\varepsilon$.
\end{enumerate}

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