Re: hint on derivative problem


jraul wrote:
Suppose f has a finite derivative in (a,b) and f(x)-->oo as x-->b from
the left. Prove that the lim (f)'(x) does not exist or is infinite as
x-->b from the left.

As an example, I'm thinking of something like f(x) = 1/x, then the
derivative tends to -infinity as x goes to zero.

This isn't an example of what you're trying to prove, since f(x) ->
-infinity as x approaches 0 from the left. A better example is f(x) =
-1/x, where b = 0.

Or some situation
where the derivative is oscillating real fast at a point (e.g.,
sin(1/x)) so that the limit of the derivative does not exist.

But can someone give me a hint on how to prove this? It is pretty
abstract in that I don't have anything concrete to work with (e.g., to
find two subsequences converging to different limits to show the limit
does not exist).

I'd try contradiction here; if the conclusion is false, then lim(f'(x))
exists as x approaches b from the left. It looks like something along
the lines of the Mean Value Theorem is required here.

--- Christopher Heckman

.

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