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\noindent {\bf Problem statement} Suppose $f(x)=x^{-\ln x}$.

\medskip

\noindent a) Verify that $\lim\limits_{x\to 0^+}f(x)=0$ and
$\lim\limits_{x\to \infty}f(x)=0$. Graph $f$ on the interval
$[0,10]$. 

\medskip

\noindent b) A remarkable result of third semester calculus is that
${\int_{-\infty}^{\infty}{e^{-x^2}}\,dx = \sqrt{\pi}}$. Assume that
this result is correct, and use it to show that
${\int_0^{\infty}{x^{-\ln\,x}}\,dx = e^{1/4}}\sqrt{\pi}$. ({\tt Maple}
{\it can}\/ ``do'' the first integral, but not the second!)

\medskip

\noindent {\bf Hint} Make a change of variables, and then complete the
square.

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\noindent c) Include a graph of $e^{-x^2}$ on the interval
$[-2,2]$. Use your answer to b) to compare this graph to the graph in
a).

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