\input homework_macros.tex

%\distiller
\printer

\def\Lim#1{\displaystyle\lim_{#1\to\infty}}
\def\dd#1#2{\frac{d#1}{d#2}}

\SN{6.7}
\pn1. Any power of $x$ grows more slowly that $a^x$ for any $a>1$.
For any $a>1$, $\log_a x$ grows more slowly than any power of $x$.
Hence a) and c) grow more slowly than $e^x$;
$\log_{10}x$ grows more slowly than $x$ and hence more
slowly than $e^x$.
Since $\sin x$ is bounded $x^2$ and $x^2+\sin^2 x$
grow at the same rate.
Hence $x^2+\sin^2 x$ grows more slowly than $e^x$.
Both $e^x$ and $e^x/2$ grow at the same rate.
Finally, if$a>b>1$ $a^x$ grows faster than $b^x$.
Since $3/2<e$, $(3/2)^x$ grows more slowly than $e^x$.
Since $4>e$, $4^x$ grows faster than $e^x$.
Since $\sqrt e<e$, $e^{x/2}$ grows more slowly than $e^x$.

\pn3.
$x^2+4x$, $\sqrt{x^4x^3}$, $(x+3)^2$, $8x^2$ and $x^2$ all 
grow at the same rate.
$x^5-x^2$ grows faster than $x^2$.
$2^x$ grows faster than any power of $x$, hence faster than $x^2$.
$x^3e^{-x}$ goes to $0$ as $x\to\infty$ so it geows more slowly
than $x^2$.
Finally, $x$ grows faster than $\ln x$ so $x^2$ grows faster than
$x\ln x$.

\pn5.
$\ln x$ and $\log_a x$ grow at the same rate.
Hence
$\log_3x$, $\ln2x=\ln2+\ln x$, $\ln\sqrt x=\frac 12\ln x$,
$5\ln x$ and $\ln x$ all grow at the same rate.
$\ln x$ grow more slowly that $x$ and $\sqrt x=x^{1/2}$.
$\ln x$ grows more slowly than $e^x$.
Finally $\frac 1x$ goes to $0$ as $x\to\infty$ and so grows
more slowly than $\ln x$.

\pn7.
$e^x$ and $e^{x/2}$ grow at the same rate;
since $e<4$, $e^x$ grows more slowly than $4^x$
which is less than both $x^x$ and $(\ln x)^x$,
so $e^x$ grows more slowly than $(\ln x)^x$.
Finally, $(\ln x)^x$ grows more slowly than $x^x$ since
$\Lim_{x}\frac{x^x}{(\ln x)^x}=\infty$.

\pn9.
a) $\Lim{x}\frac xx=1$ so false,
b) $\Lim x\frac{x}{x+5}=1$ so false,
c) $\frac x{x+5}<1$ for large $x$ so true,
d) $\frac x{2x}=\frac 12$, so true,
e) $\Lim x\frac{e^x}{e^{2x}}=\Lim x\frac 1{e^x}=0$ so true,
f) $\frac {x+\ln x} x$: since $\Lim x\frac{\ln x}x=0$, for all
large $x$, $\frac {\ln x}x<1$ so $\frac {x+\ln x} x<2$ for all
large $x$, so  $x+\ln x=O(x)$.
g) $\Lim x\frac{\ln x}{\ln(2x)}=1$ so $\ln x\neq o(\ln(2x))$.
h) $\Lim x\frac{\sqrt{x^2+5}}x=1$ so for all sufficiently large $x$,
$\frac{\sqrt{x^2+5}}x<2$, so $\sqrt{x^2+5}=O(x)$.

\pn23.\hskip-5pt a)
$\log_2 n$ grows more slowly than any power of $n$,
so $n\log_2 n$ grows more slowly than $n^{3/2}$.
Similarly, one can see $n(\log_2 n)^2$ grow more slowly
than $n^{3/2}$.
Finally, $\log_2 n$ grow more slowly than $(\log_2 n)^2$,
so $n \log_2 n$ grows the most slowly of the three given
functions.

\end

6.7  p. 503:   1,  3,  5,  7,  9, 23a)