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\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 3}
\date{Due on Moodle, September 28, 2020}
\maketitle

\begin{enumerate}

\item Compute the following limits:
\begin{enumerate}
\item $\displaystyle \lim_{\theta \to 0} \frac{\theta}{\sin \theta}$
\item $\displaystyle \lim_{\phi \to 0} \frac{1-2 \, \cos^2 \phi}\phi$
\end{enumerate}
\emph{Hint:} For (b), use the double-angle identity
\[
  \cos^2 \phi = \frac{1+\cos 2 \phi}2 \,.
\]

\item The (natural) exponential function can be defined as
\[
  \exp (x) = \lim_{n \to \infty} \biggl( 1 + \frac{x}n \biggr)^n \,.
  \tag{*}
\]
Use this definition to show that $\exp (a+b) = \exp (a) \cdot \exp (b)$
for all $a, b \in \R$.

\emph{Hint:} Convert the definition (*) into a different limit by
setting $y = x/n$.  

\item Show that the equation
\[
  x = \cos x
\]
has at least one solution in the interval $(0,\pi/2)$.

\emph{Hint:} Intermediate value theorem.

\item Let $f \colon \R \to \R$ be continuous and $g \colon \R \to \R$
some real-valued function.  Show, by giving a counter example, that
the following statement is not generally true: \emph{If
$h(x) = f(g(x))$ is continuous, then $g$ is also continuous.}

\item Compute the derivative of the following functions directly from
the definition
\[
  f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}h \,.
\]

\begin{enumerate}
\item $\displaystyle f(x) = x^2$
\item $\displaystyle f(x) = \sqrt x$
\end{enumerate}





\end{enumerate}

\end{document}
