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In first-year calculus, we define intervals such as
$(u, v)$ and $(u, \infty)$. Such an interval is a
\emph{neighborhood} of $a$
if $a$ is in the interval. Students should
realize that $\infty$ is only a
symbol, not a number. This is important since
we soon introduce concepts
such as $\lim_{x \to \infty} f(x)$.
When we introduce the derivative,
\[
\lim_{x \to a} \frac{f(x) - f(a)}{x - a},
\]
we assume that the function is defined and continuous
in a neighborhood of $a$.
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