A neat 1^infinity indeterminate example

[Dave L. Renfro]

If anyone's interested, here's a neat example of
a 1^infinity indeterminate form that I came up with
yesterday. I'm posting it because my last few posts
in sci.math have been pretty lame.

Let f(x) be any "sufficiently nice" function such
that f(1) is not zero. Any arithmetic/composition
combination of polynomials, trig. functions, exponential
functions, etc. will do for f(x), such as:

f(x) = x

f(x) = sin(x)

f(x) = ln(x+1)

Then the limit as x --> 1 of P^Q, where

P = [(x+1)*f(x)] / [2*f(1)]

Q = 1 / (x-1)

is equal to exp{ 1/2 + f'(1)/f(1) }.

To get this, apply L'Hopital's rule to
ln(P^Q) = Q*ln(P).

Dave L. Renfro

[The World Wide Wade]

In article <1187292236.8...@q3g2000prf.googlegroups.com>,

This follows from: Suppose g(1) = 1 and g'(1) exists. Then lim_{x->1}
g(x)^(1/(x-1)) = e^(g'(1)). Proof: ln(g(x))/(x-1) = [ln(g(x)) -
ln(g(1))]/(x-1) -> derivative of ln(g(x)) at 1. By the chain rule,
this is g'(1)/g(1) = g'(1). Exponentiating gives the result.

[Dave L. Renfro]

The World Wide Wade wrote:

> This follows from: Suppose g(1) = 1 and g'(1) exists. Then lim_{x->1}
> g(x)^(1/(x-1)) = e^(g'(1)). Proof: ln(g(x))/(x-1) = [ln(g(x)) -
> ln(g(1))]/(x-1) -> derivative of ln(g(x)) at 1. By the chain rule,
> this is g'(1)/g(1) = g'(1). Exponentiating gives the result.

Nice argument. I saw limit x --> 1 of {(x/2)(x+1)}^{1/(x-1)} in the
exercises of an old calculus book (about 100 years old) yesterday
morning while looking at something else (classical curve sketching
methods for graphs of implicitly defined "functions", if you must
know), and while working it out via logarithms and L'Hopital's rule,
I noticed it easily generalized to what I posted. Although I didn't
try to evaluate it using another approach, I doubt I would have
come up with what you did if I had tried. I only posted it because
I happened to have a little free time, I have not posted much lately,
and I didn't feel like launching into one of my extended essays on
some topic.

Dave L. Renfro