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%	\mbox{}
%	\title{Limits and Derivatives (MATH 53)}
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%	\maketitle
	
	
\section*{Limits and Continuity}
\bigskip

\begin{problem}[jS, Exercises~14.2(10,13,16,19,22)]
	Find the limit, if it exists, or show that the limit does not exists.
	\begin{enumerate}
		\item $\lim_{(x,y) \to (0,0)} \frac{5y^4 \cos^2 x}{\sqrt{x^4 + y^4}}$.
		\item $\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}}$.
		\item $\lim_{(x,y) \to (0,0)} \frac{xy^4}{x^4 + y^4}$.
		\item $\lim_{(x,y,z) \to (\pi,0,1/3)} e^{y^2} \tan(xz)$.
		\item $\lim_{(x,y,z) \to (0,0,0)} \frac{x^2 y^2 z^2}{x^2 + y^2 + z^2}$.
	\end{enumerate}
\end{problem}

\begin{problem}[jS, Exercises~14.2(39,40,41)]
	Use polar coordinate to find the following limits.
	\begin{enumerate}
		\item $\lim_{(x,y) \to (0,0)} \frac{x^3 + y^3}{x^2 + y^2}$.
		\item $\lim_{(x,y) \to (0,0)} (x^2 + y^2) \ln(x^2 + y^2)$.
		\item $\lim_{(x,y) \to (0,0)} \frac{e^{-x^2 -y^2} - 1}{x^2 + y^2}$.
	\end{enumerate}
\end{problem}

\begin{problem}[jS, Exercises~14.2(30,35,36)]
	Determine the set of points at which the following functions are continuous.
	\begin{enumerate}
		\item $f(x,y) = \cos(\sqrt{1 + x - y})$.
		\item $f(x,y,z) = \arcsin(x^2 + y^2 + z^2)$.
		\item $f(x,y,z) = \sqrt{y - x^2} \ln z$.
	\end{enumerate}
\end{problem}

\begin{problem}[jS, Exercise~14.2.45]
	Show that the function $f(x) = |x|$ is continuous in $\rr^n$. [Hint: Consider $|x - a|^2 = (x-a)\cdot(x-a)$.]
\end{problem}

\vspace{20pt}
\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}