Re: sin x / x tends to 1...

>>> Then please define what the "x" in "sin(x) / x" is supposed to mean.
>>
>> It's a real variable.

I think a *lot* of the confusion and acrimony in this thread is due
to a lack (one I've contributed to) of *explicit* "dimensional analysis"
in the sense of physics.

If we develop enough "elementary theory of area", supplemented by
very simple limit arguments, to make (the usual) sense of "the area
of a circular sector", then we can--and many have be tempted to--
define sin(x) to be the vertical cartesian coordinate (I never
remember if that's the abscissa or the ordinate) of the point
on the unit circle such that the sector bounded by that radius
and the radius on the positive horizontal cartesian axis
has area x/2. That leads, however, to the problem that when
we form the quotient sin(x)/x whose limit at 0 we wish to evaluate,
we are dividing something with dimension "length" (namely, sin(x))
by something with dimension "length^1" (namely, twice the area
of the sector). This is easy to remedy: let sin(x) be, not the
length described above, but the area of the rectangle with that
length as its height and the (unit length) radius on the positive
horizontal axis as its width. Now it's really, really easy to
see that the sector in question is *nearly* the same as the
half of that rectangle below its diagonal starting at the origin,
and nearly as easy to prove (using no more geometry than the
usual formula for the area of a triangle, no more about area
than that it is additive in the appropriate sense, and very
little algebraic manipulation) that the desired limit exists
and equals the dimensionless number 1.

HOWEVER, this is unsatisfactory (to some) because they want
(without saying so explicitly) x to have the dimension of
length, not length^2, so that (among other presumed benefits)
they can leave sin(x) as itself a length and still get a
dimensionless quotient. It is at this point in the argument
that I proposed, without being clear about it even to myself,
that this can be done by applying the (assumed accepted, as
above) theory of area for circles and circular sectors, plus
a *geometric* interpretation of the derivative of the radial
variation of such areas, to *derive* a theory of arclength
for circles and circular sectors. Well, in fact, this
proposal respects the dimensional analysis very nicely.
So I'll stand by it with more confidence.

People who want to eliminate the dimensions from the problem,
and use dimensionless numbers from the get-go, probably should
stick with the power series or ODE methods, and only return to
the geometry after they've done the analysis.

Lee Rudolph
.

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