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\vspace*{.3in}
\noindent{\bf Beals, page 22 \#9:}
No, it's not always true.  Consider for example the function $f:\R\to\R$
which is zero everywhere except on intervals of the form $[n-1/n^2,n+1/n^2]$,
$n\in\N$; and on such an interval the graph of $f$ is the pair of lines
joining the points $(n-1/n^2,0)$ and $(n+1/n^2,0)$ to the point $(n,1)$.
Then $f$ is continuous and $f(n) = 1$ for every $n\in\N$, so 
$\lim_{x\to\infty} f(x)\neq 0$ (in fact, the limit doesn't exist).  However, 
$$
\int f = \sum_{n=1}^\infty \frac{1}{n^2} < \infty.
$$
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