VI$.$

$1.$

Prove that if $f(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt,$ then $f=0.$

$2.$

Find all continuous functions $f$ satisfying

$($i$){\displaystyle \underset{0}{\overset{x}{}}}f=e^x.$

$($ii$){\displaystyle \underset{0}{\overset{x^2}{}}}f=1-e^{2x^2}.$

$3.$

$($a$)$ Prove that

$1x\frac{x^2}{2!}\frac{x^3}{3!}$ cdots $\frac{x^n}{n!}{e}^x,$ for $x0$

Hint: Use induction on $n,$ and compare derivatives.

$($b$)$ Give a new proof that $\underset{x}{lim}\frac{e^x}{x^n}=.$

$4.$

$($a$)$ Find $\underset{y0}{lim}log\frac{1y}{y}.($You can use l$\text{'}$H$$pital$\text{'}$s Rule, but that would be silly.$)$

$($b$)$ Find $\underset{x}{lim}xlog(1\frac{1}{x}).$

$($c$)$ Prove that $e=\underset{x}{lim}(1\frac{1}{x}{)}^x.$

$($d$)$ Prove that $e^a=\underset{x}{lim}(1\frac{a}{x}{)}^x.($It is possible to derive this from part $($c$)$ with just a little algebraic fiddling.$)$