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%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
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\begin{document}
	
	\mbox{}
	\title{Problem Set 9: Mean Value Theorem and L'Hopital's Rule}
	
	\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}
		Show that the following equations have exactly one root:
		\[
			2x + \cos x = 0 \quad \text{ and } \quad x^3 + e^x = 0.
		\]
	\end{problem}

	\begin{problem}
		Show that a polynomial of degree $3$ has at most three real roots.
	\end{problem}


	\begin{problem}
		Does there exist a function $f$ such that $f(0) = -1$, $f(2) = 4$, and $f'(x) \le 2$ for all $x$?
	\end{problem}
	
	\begin{problem}
		Show that $\sin x < x$ if $0 < x < 2\pi$.
	\end{problem}

	\begin{problem}
		Argue that $|\sin a - \sin b| \le |a - b|$ for all $a,b \in \rr$.
	\end{problem}
	
	\begin{problem}
		Prove that if $f$ is a differentiable function such that $f'(x) \neq 1$ for all real numbers $x$, then there exists at most one $x \in \rr$ such that $f(x) = x$.
	\end{problem}

	\begin{problem}
		Find the following limits.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to \infty} \frac{e^{x/10}}{x^3}$;
			\item $\lim_{x \to \infty} \frac{(\ln x)^2}{x}$;
			\item $\lim_{x \to 0} \frac{x + \sin x}{x + \cos x}$;
			\item $\lim_{x \to \infty} x \tan(1/x)$;
			\item $\lim_{x \to \infty} x^{1/x}$;
			\item $\lim_{x \to 0} (\csc x - \cot x)$;
			\item $\lim_{x \to \infty} \bigg(\frac{2x-3}{2x+5}\bigg)^{2x+1}$.
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		 Evaluate
		 \[
		 	\lim_{x \to \infty} \bigg[x - x^2 \ln\big(\frac{1+x}{x}\big)\bigg].
		 \]
	\end{problem}

	\begin{problem}
		Suppose that $f$ is a positive function such that $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = \infty$. Show that $\lim_{x \to a} f(x)^{g(x)} = 0$.
	\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}