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From: Fred Nurk on 10 Jul 2010 18:18
1. How do you get from lim h->0 f(x + h) / h to lim h->0 f(x + h) in the
second-last to last line found in my screenshot of the proof of the
product rule at the bottom of http://sites.google.com/site/xtheunknown0/
maths/differentiation?

2. How are you supposed to write those expressions in computer notation?

TIA,
Fred
From: Arturo Magidin on 10 Jul 2010 19:29
On Jul 10, 5:18 pm, Fred Nurk <albert.xtheunkno...(a)gmail.com> wrote:
> 1. How do you get from lim h->0 f(x + h) / h to lim h->0 f(x + h) in the
> second-last to last line found in my screenshot of the proof of the
> product rule at the bottom ofhttp://sites.google.com/site/xtheunknown0/
> maths/differentiation?

You should really stop posting links and start writing out what you
want addressed; I for one am rather wary of clicking on links posted
in public forums. Moreover, the links you give are pretty lousy in
terms of giving the reader access to what you want. They are full of
useless information since they are mainly screenshots. It ends up
seeming like you think your time is too precious to spend making your
question easily understandable/accessible, and you would rather have
your readers spend theirs in helping you.

Finally, this particular screenshot has no proof of the product rule
that I can find, so you've wasted my time.


> 2. How are you supposed to write those expressions in computer notation?


You mean, how do you type it out in ASCII? Pseudo-TeX is the standard
way of doing it here. A quick Google Search revealed

http://mathforum.org/typesetting/ascii.guidelines.html

(disclaimer: try to avoid using ASCII art, like their answers (a) for
fractions and exponents; they don't always render correctly in other
people's viewers).

As to how to write your specific question, since you failed to provide
it, either in-message or in the link you gave, I have no idea.

--
Arturo Magidin
From: Arturo Magidin on 10 Jul 2010 19:50
On Jul 10, 6:29 pm, Arturo Magidin <magi...(a)member.ams.org> wrote:
> On Jul 10, 5:18 pm, Fred Nurk <albert.xtheunkno...(a)gmail.com> wrote:
>
> > 1. How do you get from lim h->0 f(x + h) / h to lim h->0 f(x + h) in the
> > second-last to last line found in my screenshot of the proof of the
> > product rule at the bottom ofhttp://sites.google.com/site/xtheunknown0/
> > maths/differentiation?

[...]

> Finally, this particular screenshot has no proof of the product rule
> that I can find, so you've wasted my time.

Ah; google broke your link. Yet another reason to avoid posting links
to screenshots and instead posting what you mean.

The section in question reads:

F'(x) = lim_{h->0) (F(x+h)-F(x))/h
= lim_(h->0) ( f(x+h)g(x+h) - f(x)g(x) )/h

Adding and subtracting f(x+h)g(x),

F'(x) = lim_(h->0) ( [f(x+h)g(x+h) - f(x)g(x)] + [f(x+h)g(x) - f(x
+h)g(x)] )/h

Regrouping:

F'(x) = lim_{h->0) ( [f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) -
f(x)g(x)] )/ h
= lim_{h->0) ( f(x+h)[g(x+h)-g(x)] + g(x)[f(x+h)-f(x)] )/h
= (lim_{h->0) f(x+h)) * (lim_{h->0} (g(x+h)-g(x))/h))
* (lim_{h->0} g(x)) * ( lim_{h->0}( (f(x+h)-f(x))/h ))
---

You are incorrect. They did NOT "go from lim_(h->0)( f(x+h)/h) to
lim_{h->0)f(x+h).

Remember that (a*b)/c = a*(b/c). For example, 6/5 is the same as
2*(3/5) and as 3*(2/5); it is *not* the same as (2/5)*(3/5).

In short, you had an expression of the form

(AB + CD)/h

(where A = f(x+h), B=g(x+h)-g(x), C=g(x), D=f(x+h)-f(x) ).

Using algebra, you have

(AB +CD)/h = ( (AB)/h ) + ((CD)/h)
= A*(B/h) + C*(D/h).

That's all they did.

Then you use the fact that the limit of a product is the product of
the limits (provided the limits in question exist). The same is true
in the second part.

--
Arturo Magidin

PS. That's how you "are supposed to write those expressions" in the
newsgroup so your readers don't have to spend *their* time trying to
figure out your question.
From: David C. Ullrich on 11 Jul 2010 08:07
On Sat, 10 Jul 2010 22:18:06 GMT, Fred Nurk
<albert.xtheunknown0(a)gmail.com> wrote:

>1. How do you get from lim h->0 f(x + h) / h to lim h->0 f(x + h) in the
>second-last to last line found in my screenshot of the proof of the
>product rule at the bottom of http://sites.google.com/site/xtheunknown0/
>maths/differentiation?
>
>2. How are you supposed to write those expressions in computer notation?

Basic rule: _Read_ a few posts in a group before posting.
The way you're supposed to write those expressions here
is the same as the way everyone else does it.

>
>TIA,
>Fred

From: Fred Nurk on 12 Jul 2010 05:15
Arturo Magidin wrote:
> <snip>
> You are incorrect. They did NOT "go from lim_(h->0)( f(x+h)/h) to
> lim_{h->0)f(x+h).
> <snip>

Ah, I see now. Thank you.
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