🚀 go-pugleaf

Sheila.King@f315.n103.z1.fidonet.org (Sheila King) writes:
>So, what you're saying is that any time a sum or difference involves a
>divergent series, there is no answer to the sum or difference? Is this
>true in all cases?

Right -- this is the *definition* of "divergent"!

>But there are techniques for evaluating limits that tend to infinity
>minus infinity. These are commonly taught in first year calculus in the
>chapter including L'Hopitals rule. How are these different from this
>example?

Those are limits of functions that are well-behaved except at one
point.  For example, if you take lim [x->0] (sin x)/x then you are
looking at a situation in which (sin 0.0001)/0.0001 is well-defined,
and all those values approach a limit.  And even then, you can't say
that the limit *is* the value of (sin 0)/0.  That expression has no
value.

In the original question, the integral is the lim [delta x -> 0] of a
sum of areas of tall skinny rectangles.  Each finite value of delta x
already is problematic, because each includes one or two rectangles
right at the y axis that are infinitely tall.

The best you can say is that 

	lim [a -> 0] <integral from -2 to -a> + <integral from a to 2>

is zero.  That's true; as a approaches zero the two integrals have larger
and larger magnitudes, but their sum is still zero for any nonzero a.