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\begin{document}

\vspace*{-.7cm}
\parbox[b]{1.65in}{\Huge MATH 226} \
\parbox[b]{2in}{\normalsize {Assignment 1 \\[-3pt]
April 16, 2020}} \ \hfill Name \rule{2in}{.5pt}

\normalsize


\begin{problem}
There is a simple formula expressing the sign function in terms of
the unit step function. Can you discover this formula? Remember
\[
\Sign(x) : = \begin{cases} - 1 &   \text{if} \ \  x < 0  \\
 \phantom{-} 0 &    \text{if} \ \ x = 0  \\
\phantom{-} 1 &  \text{if} \ \ x > 0  \\
\end{cases}
\qquad \text{and}  \qquad \us(x) : = \begin{cases} 0 &
    \text{if} \ \  x < 0  \\
 1 &    \text{if} \ \ x \geq  0.
\end{cases}
\]
We are seeking an explicit closed form formula, not a piecewise formula. Prove the formula that you discovered.
\end{problem}

\begin{problem}
Prove that for all real numbers $x$ and $y$ we have
\[
\max\{x,y\} = \frac{1}{2} \bigl(x+y + |x-y|\bigr).
\]
Discover and prove the analogous formula for the minimum.
\end{problem}

\begin{problem}
\begin{enumerate}
\renewcommand*\theenumi{\Roman{enumi}}
\renewcommand*\labelenumi{(\theenumi)}

\item
Consider the function
\[
f(x) = \bigl| \Sign(x)  \bigr|, \qquad x\in ??.
\]
\begin{enumerate}[(a)]
 \item
Determine the natural domain and the range of this function.
\item
Sketch a detailed graph of this function. Follow the logic of the graphs of $\Sign$, $\us$, $\Floor$ and $\Ceiling$ and mark some points by disks ($\bullet$) and some by circles ($\circ$).
\item
Write a clear mathematical sentence in which you will state at which points you placed disks and at which points you placed  circles.
\end{enumerate}

\item
Consider the function
\[
f(x) = \Sign(\sin(\pi x)), \qquad x \in ??.
\]
\begin{enumerate}[(a)]
 \item
Determine the natural domain and the range of this function.
\item
Sketch a detailed graph of this function. Follow the logic of the graphs of $\Sign$, $\us$, $\Floor$ and $\Ceiling$ and mark some points by disks ($\bullet$) and some by circles ($\circ$).
\item
Write a clear mathematical sentence in which you will state \underline{all} the points at which you placed disks and \underline{all} the points at which you placed  circles.
\end{enumerate}
\end{enumerate}
\end{problem}


\begin{problem}
Let  $x, y, z \in  \nR$. Prove the following inequalities
\begin{enumerate}[(a)]
\item
$\displaystyle  |x  -  y|  \leq |x - z | + |z-
y|$.

\item
$\displaystyle \bigl| |x| - |y| \bigr| \leq |x -
y|$.
\end{enumerate}
\end{problem}

\begin{problem}
Use the definition of limit to prove that
\[
\lim_{x\rightarrow +\infty}  \ \frac{(\sin x)^2}{x\, (\sin x)^2 + 1}
= \ \ ? .
\]
\end{problem}

\begin{problem}
Use the definition of limit to prove that
\[
\lim_{x\rightarrow +\infty}  \ \frac{\sin x}{x\, (\sin x)^2 + 1} = \
\ ? .
\]
\end{problem}


\begin{problem}
Determine whether the function
\[
x \mapsto  \frac{1}{x\, (\sin x)^2 + 1}
\]
has a limit when $x \rightarrow +\infty$.  State your conclusion and
justify it.
\end{problem}

\end{document}
