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\begin{center}Math 335 Real Analysis

Exam I

September 21, 2001

\end{center}

\textbf{Answer all 5 questions.  All questions have equal points. 
Symbols and markings without complete sentences will not be considered as
answers.}

\begin{enumerate}
\item Let $f$ be a function defined on \textbf{R} and $L$ a real number.
\begin{itemize}

\item[{a)}] Define what is meant by $f (x) \rightarrow L $ as $x
\rightarrow a.
$

\item[{b)}] Define what is meant by saying $f$ is continuous at $x = a$.
\end{itemize}

Let $f (x) =  x^2 + 1. $

Find $\delta$ so that $\mid f (x) - f (o) \mid < \epsilon $ when $\mid x
- o \mid < \delta.  $  Your answer will give $\delta $ in terms of
$\epsilon$.

\vbox to 5.5in{}
\pagebreak

\item Suppose $f_1$ and $f_2$ are two functions defined on \textbf{R} and
$\lim\limits_{x \to a} f_1 (x) = L_1 $ and $ \lim\limits_{x \to a} f_2 (x)
= L_2.  $ Define $ g (x) = f_1 (x) + f_2 (x).  $  Prove that $
\lim\limits_{x
\to a} g (x) = L_1 + L_2.$
 
\pagebreak
\item Let a $>$ -1. Use induction to prove  $(1 + a)^n \geq 1 + na $
for all positive integers $n$.

\pagebreak
\item Determine each of the following limits:
\begin{itemize}
\item[{i)}] $\lim\limits_{n \to + \infty}  (3 + \cos n) n$

\item[{ii)}]$\lim\limits_{x \to 2^-} \frac{\sqrt{4 - x^2}}{\sqrt{6 - 5x +
x^2}}$

\item[{iii)}]$\lim\limits_{x \to 0} \frac{\sin^2 x}{x^2} $
\end{itemize}

You must give the basis of your determination in each case.

\pagebreak
\item Let $f$ be a function on $\mathbb{R}$ defined by 
\vskip.15in

$ f(x) = \left\{ \begin{array}{ll}
\frac{2 (x^2 - 9)}{x - 3} & x \neq 3 \\
12 & x = 3
\end{array}
\right. $

Show that $\mid f(x) - f(3) \mid < \epsilon $ when $\mid x - 3 \mid <
\frac{\epsilon}{2}, $ for all $ \epsilon > 0.
$

Is the function of continuous at $x = 3 $?  Yes or No.

\end{enumerate}



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