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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{Friday, December 2, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 6}
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\begin{document}

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\begin{questions}
    
%%Fitzpatrick 3.7.11
\question Let $k$ be a natural number.  Prove that $\displaystyle \lim_{x \to 1} \frac{x^k - 1}{x - 1} = k$.

%%Fitzpatrick 3.5.8
\question Suppose that a continuous function \ $f : \mathbb{R} \to \mathbb{R}$ \ is periodic; that is, there is a number \ $p > 0$ \ such that \ $f(x + p) = f(x)$ \ for all $x \in \mathbb{R}$. \ Show that \ $f : \mathbb{R} \to \mathbb{R}$ \ is uniformly continuous.

%%Fitzpatrick 3.4.11
\question A function \ $f : D \to \mathbb{R}$ \ is said to be \emph{Lipschitz} provided that there is a number \ $C \geq 0$ \ with
\begin{displaymath} |f(u) - f(v)| \leq C \, |u - v| \quad \text{for all $u$ and $v$ in $D$}. \end{displaymath}
Show that if a function $f$ is a Lipschitz function on $D$, then it is \emph{uniformly} continuous on $D$.

%%19.4
\question Prove that if $f$ is uniformly continuous on a bounded set $S$, then $f$ is a bounded function on~$S$.

%%19.7
\question Let $f$ be a continuous function on $[0, \, \infty)$.  Prove that if $f$ is uniformly continuous on $[k, \, \infty)$ for some $k > 0$, then $f$ is uniformly continuous on $[0, \, \infty)$.

%%19.12
\question Let $f$ be a continuous function on $[a, \, b]$.  Show that the function $f^*$ defined by
\begin{displaymath} f^*(x) = \sup \left\{f(y) \mid a \leq y \leq x \right\}, \text{ for } x \in [a, \, b], \end{displaymath}
is an increasing continuous function on $[a, \, b]$.

%%20.16
\question Suppose the limits \ $\displaystyle L_1 = \lim_{x \to a^+} f_1(x)$ \ and \ $\displaystyle L_2 = \lim_{x \to a^+} f_2(x)$ \ both exist.
\begin{parts}
\part Show that if \ $f_1(x) \leq f_2(x)$ \ for all $x$ in some interval \ $(a, \, b)$, \ then \ $L_1 \leq L_2$.
\part Suppose \ $f_1(x) < f_2(x)$ \ for all $x$ in some interval \ $(a, \, b)$. \ Can you conclude that \ $L_1 < L_2$?
\end{parts}

%%20.17
\question Show that if \ $\displaystyle \lim_{x \to a^+} f_1(x) = \lim_{x \to a^+} f_3(x) = L$ \ and if \ $f_1(x) \leq f_2(x) \leq f_3(x)$ \ for all $x$ in some

interval \ $(a, \, b)$, \ then \ $\displaystyle \lim_{x \to a^+} f_2(x) = L$.  (Note: Be sure to show that \ $\displaystyle \lim_{x \to a^+} f_2(x)$ \ exists, since this is not assumed.)

%%20.18
\question Let \ $\displaystyle f(x) = \frac{\sqrt{1 + 3x^2} - 1}{x^2}$ \ for $x \not= 0$. \ Show that \ $\displaystyle \lim_{x \to 0} f(x)$ \ exists and determine its value.

Be sure to justify all claims.

%%20.20
\question Let $f_1$ and $f_2$ be functions such that \ $\displaystyle \lim_{x \to a^S} f_1(x) = +\infty$ \ and such that the limit \ $\displaystyle L_2 = \lim_{x \to a^S} f_2(x)$ \ exists.
\begin{parts}
\part Prove that \ $\displaystyle \lim_{x \to a^S} \left(f_1 + f_2\right)(x) = +\infty$ \ if \ $L_2 \not= -\infty$.
\part Prove that \ $\displaystyle \lim_{x \to a^S} \left(f_1 \, f_2\right)(x) = +\infty$ \ if \ $0 < L_2 \leq +\infty$.
\part Prove that \ $\displaystyle \lim_{x \to a^S} \left(f_1 \, f_2\right)(x) = -\infty$ \ if \ $-\infty \leq L_2 < 0$.
\part What can you say about \ $\lim_{x \to a^S} \left(f_1 \, f_2\right)(x)$ \ if \ $L_2 = 0$?
\end{parts}

\end{questions}

\end{document}

