Proving Limit Property: Easy Steps

[andilus]

how to prove
lim(x->a)\frac{f(x)-f(a)}{x-a}=lim(h->0)\frac{f(a+h)-f(a)}{h}

it seems to be obvious, but i don't know how to prove```

 

[statdad]

What is the relationship between the denominator x - a and the denominator h?

 

[andilus]

What is the relationship between the denominator x - a and the denominator h?

Sorry,I have not express clearly.
what i want to prove is just:
lim(x->a)(\frac{f(x)-f(a)}{x-a})=lim(h->0)(\frac{f(a+h)-f(a)}{h})

 

[l'Hôpital]

Listen to what statdad is saying. What is the relationship by h and x - a? The answer to this question will essentially answer your question.

 

[wisvuze]

use a delta-epsilon argument, the standard delta epsilon definition:

as x approaches a, we have:

given \epsilon > 0, there exists a \delta >0 such that for all x with the property 0 < | x - a | < \delta, then |f(x) - L | < \epsilon.

In this argument, we have the distance between a point x and a fixed point a bounded between 0 and some fixed \delta. Can you provide a similar argument as h approaches ____ ?

 

[Bohrok]

Why not let h = x - a and rewrite the limit after the substitutions?

 

[HallsofIvy]

That is exactly what statdad was suggesting!

 

[Bohrok]

I wasn't sure that that was what statdad was getting at, which is why I posted the equation. Some things are just too subtle, at least for me.

 

[HallsofIvy]

You still don't get it? In the first limit,
\lim_{x\to a}\frac{f(x)- f(a)}{x-a}

let h= x- a. Then x= ??

 

[Mark44]

use a delta-epsilon argument, the standard delta epsilon definition:

as x approaches a, we have:

given \epsilon > 0, there exists a \delta >0 such that for all x with the property 0 < | x - a | < \delta, then |f(x) - L | < \epsilon.

In this argument, we have the distance between a point x and a fixed point a bounded between 0 and some fixed \delta. Can you provide a similar argument as h approaches ____ ?

I'm pretty sure none of this is applicable to the problem in this thread.

 

[Bohrok]

No, I understand perfectly how to use h = x - a to transform the first to the second; I just didn't see that statdad was hinting at doing it that way.

 

[statdad]

Sorry for any confusion I caused. I've always believed the best horror movies ( and books) are the ones that hint at the source of the horror, and that the best hints are ones that make you puzzle out their meaning. This time, apparently, I was a little too vague.