Q&A #6003

2nd semester Calculus

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From: Joshua (for Teacher2Teacher Service)
Date: Apr 02, 2001 at 16:49:13
Subject: Re: 2nd semester Calculus

Hi Kerry,
When I teach L'Hopital's rule, I try to base it on things that
they've already learned about local linear approximations and/or
Taylor series.

For example, why is the limit x->0 of (sin x)/x equal to zero?
Well, we used that fact already to prove that the derivative of
sin x is cos x, so it's really circular reasoning to use the
derivative of sin x in L'Hopital's rule to prove this limit, but
it will illustrate the idea anyway.

Since the slope of sin x at the origin is 1, to first order in x,
sin x = x near the origin.  Since x is small, any x^2 term will
be negligible by comparison.  So, sin x = (derivative of sin x)*x,
and hence sin x / x = (derivative of sin x) = 1.

That illustrates the principle in general: if f(a) = 0, then near
a, f(x) is approximately equal to 0 + f'(a) * (x-a).  That's the
equation of the tangent line to f(x) at the point a.  That's also
first step in developing Taylor/MacLaurin series!
If g(a) = 0, g(x) is approximately equal to its tangent line,
0 + g'(a) * (x-a).

Dividing those two gives f'(a) / g'(a), which is just L'Hopital's rule.

In practice, in my physics courses people rarely used L'Hopital's rule
by name.  Instead, they used the idea:  They took the first nonzero term
of the Taylor series of both top and bottom and then divided those
to find the limit.  I'm having a hard time thinking of a specific
example of that offhand, though.

I hope that helps!  Have fun teaching it, and feel free to write
back if you'd like to continue this conversation.

 -Joshua, for the T2T service

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