Language problem

A vague notion of "limit" is:
$\lim_{x\to a} f(x) = L$ iff as $x$ gets closer to $a$, $f(x)$ gets
closer to $L$.

Unfortunately, this suggests that limits are -monotone-, which they
emphatically aren't!

A key example to bear in mind:
$1/n \to 0$,
but so does
$1, 0, 1/2, 0, 1/3, 0, 1/4, 0, \dots$

I prefer to say that a limit is like a black hole:
for each distance $\epsilon$, eventually you come within $\epsilon$
-and never escape-! (evil laugh)
What's the number for the ``eventually''? It's $N$ or $\delta$.

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A key example is the limit of $(\sin x)/x$ as $x \to \infty$:
this has limit $0$, but it never stops oscillating.
Physically, you can think about a dampened harmonic oscillator, like a
pendulum slowing down.
