Re: Question About L'Hospital's Rule in Rudin

In article
<290526.1179254622211.JavaMail.jakarta@xxxxxxxxxxxxxxx
rum.org>,
Jonathan Groves <JGroves@xxxxxxxxxx> wrote:
Dear All,

I am puzzled about Rudin's statement of L'Hospital's
Rule in his "Principles of Mathematical Analysis"
(Theorem 5.13). Here is his statement of the
theorem, and I will explain below why I am puzzled.

Suppose f and g are real and differentiable in
(a,b), and g'(x) is
not 0 for all x in (a,b), where -infinity <= a < b
<= infinity (<=
means less than or equal to). Suppose
f'(x)/g'(x)-->A as x-->a. If
f(x)-->0 and g(x)-->0 as x-->a or if g(x)-->infinity
as x-->a, then
f(x)/g(x)-->A as x-->a.

Are the limits meant to be one sided?

I forgot to say he mentions this result is true as well if x-->b or if g(x)-->-infinity as a note after the statement of the theorem.
It appears from the statement of the theorem that these limits are one-sided since x-->a and he discusses f and g only on the interval (a,b). Since he says the theorem is true as well when x-->b, it is not hard to extend this result to two-sided limits. That is, if x-->x_0 (on both sides) and f and g are real and differentiable on (a,b) containing x_0 and g' is nonzero on (a,b) except possibly at x_0, then f and g are differentiable on (a,x_0) and (x_0,b) and g' is nonzero on both these intervals, so his result applies to both one-sided limits as x-->x_0.

Why does Rudin assume g(x)-->infinity as x-->a
without assuming the same thing for
f(x)?

Because the extra assumption on f(x) is unnecessary,
thus yielding a
stronger result (fewer assumptions means the result
can be applied
more generally, and so it is considered stronger). As
stated, the
theorem could be applied to cases where f(x) has a
finite limit, has an
infinite limit, or even where f(x) has NO limit as
x-->a, if such
exist. In any case, it means that you only need to
check the limit of
g(x) before deciding whether the theorem is
applicable.

One I asked is puzzled, and another I asked says
that if
g(x)-->infinity, then f(x) is forced to approach
infinity as well. If
this is indeed the case, then I cannot see why this
is true.

It is of course not true that f(x) is "forced to
approach
infinity". Just take your favorite function that does
not, e.g.,
sin(x).

I agree. sin(x)/x^2-->0 as x-->infinity (by the squeeze theorem) and g(x)=x^2 goes to infinity, but f does not. And all the hypotheses are satisfied since both f(x)=sin x and g(x)=x^2 are differentiable, g'(x) is nonzero except at 0 (which is not a problem since we may take (a,b)=
(1, infinity) as the interval in question), and f'(x)/g'(x)=cos x/2x-->0 as x-->infinity. So I now realize all the hypotheses of the theorem can be satisified with the assumption that g(x)-->infinity but f(x) does not.


Rudin's proof, as far as I know, is correct, and he
never assumes in the proof that
f(x)-->infinity, so apparently this is not a
misprint.

Thanks for your answers.

I don't really understand what is confusing you.
Perhaps you are
thinking that if the limit of f(x) as x-->a exists
then the limit of
f/g is not an indeterminate, so why would you use
L'Hopital's Rule?
Well, that is perhaps true, but then L'Hopital's Rule
is not
restricted to being true when lim(f)/lim(g) does not
make sense.

What confused me is wondering if L'Hospital's Rule is still true if g(x)-->infinity without assuming anything about the limit of f(x) as x-->a since I have never seen L'Hospital's Rule without this assumption. Furthermore, most calculus books I've seen warn students that L'Hospital's Rule does not apply if limit of f(x)/g(x) as x-->a is not an indeterminate form. So I was thinking if you omit the hypothesis about f(x)-->infinity when g(x)--> infinity, then the theorem may no longer be true.


======================================================
================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill
Bill Watterson)
======================================================
================

Arturo Magidin
magidin-at-member-ams-org

.

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