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\begin{document}
	
	\mbox{}
	\title{Problem Set 3: Computing Limits}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
	\begin{problem}
		Find the equation of a tangent line to the graph of the function $f(x) = x^2 + 4x + 3$ passing by the point $P = (0,-2)$.
	\end{problem}

	\begin{problem}
		Use the graph of the function $f(x) = \frac{x^2 + x}{\sqrt{x^3 + x^2}}$ to state the value of each limit if it exists. If it does not exists explain way.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to 0^-} f(x)$;
			\item $\lim_{x \to 0^+} f(x)$;
			\item $\lim_{x \to 0} f(x)$.
		\end{enumerate}
		Do the same if $f(x) = (1 + e^{1/x})^{-1}$.
	\end{problem}

	\begin{problem}
		Sketch a graph of an example of a function $f$ that satisfies all of the following conditions:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to 0^-} f(x) = 2$;
			\item $\lim_{x \to 0^+} f(x) = 0$;
			\item $\lim_{x \to 4^-} f(x) = 3$;
			\item $\lim_{x \to 4^+} f(x) = \infty$;
			\item $f(0)=2$;
			\item $f(4)$ is not defined.
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Determine the infinite limits:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to 0^+} \ln(\sin x)$;
			\item $\lim_{x \to \pi^-} \cot x$;
			\item $\lim_{x \to 2^+} \frac{x^2 - 2x - 8}{x^2 - 5x + 6}$.
			\item $\lim_{x \to 0^+} \bigg(\frac 1x - \ln x \bigg)$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Evaluate each of the following limits, it it exists:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to 1} \frac{x^4 - 1}{x^3 - 1}$;
			\item $\lim_{t \to 0} \frac{\sqrt{1+t} - \sqrt{1-t}}t$;
			\item $\lim_{x \to -4} \frac{\sqrt{x^2 + 9} - 5}{x + 4}$.
			\item $\lim_{h \to 0} \frac{(x+h)^3 - x^3}h$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		If $2x \le g(x) \le x^4 - x^2 + 2$ for all $x$, evaluate $\lim_{x \to 1} g(x)$.
	\end{problem}

	\begin{problem}
		Use the Squeeze Theorem to show that
		\[
			\lim_{x \to 0} \sqrt{x^3 + x^2} \sin \frac{\pi}{x} = 0.
		\]
	\end{problem}

	\begin{problem}
		Prove that $\lim_{x \to 0} x^4 \cos \frac 2x = 0$.
	\end{problem}

	\begin{problem}
		Find each of the following limits if it exists. If it does not exist, explain why.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to -6} \frac{2x + 12}{|x+6|}$;
			%\item $\lim_{t \to 0} \frac{\sqrt{1+t} - \sqrt{1-t}}t$;
			%\item $\lim_{x \to -4} \frac{\sqrt{x^2 + 9} - 5}{x + 4}$.
			\item $\lim_{x \to 0^+} \bigg(\frac 1x - \frac 1{|x|} \bigg)$.
		\end{enumerate}
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}