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pk1234
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Hey guys, first year university math student here. I need some help explaining the proof used in the scripts I'm studying from - just part of the proof to be more precise. English isn't my first language and I don't have much experience writing/rewriting down proofs and I don't know how to write those nice latex symbols, so sorry in advance if something doesn't make sense:
Presuming:
(1), a is element of R (|a| =/= +oo)
(2), f and g are real functions
(3), limit x->a_+ (f'(x) / g'(x)) exists (must be element of R, or +-oo)
(4), limit x->a_+ (f(x)) = limit x->a_+ (g(x)) = 0
then
limit x->a_+ (f(x))/(g(x)) = limit x->a_+ (f'(x))/(g'(x))
I think I understand most of the proof but there's something right at the start that I'm completely stuck at and still don't understand precisely enough:
Let L=limit x->a_+ (f'(x) / g'(x)).
There exists delta>0, such that for all x element of (a,a+delta), f and g are both defined on this interval,
- I think this can be proved easily from (4), correct? Also, |f| and |g| are both smaller than some Epsilon>0. The following however, I don't understand at all:
and both f' and g' have a finite (not = oo or -oo) derivation on this interval, and also g'=/=0.
Why is the derivation necessarily finite?
EDIT:
To explain where I see the problem a bit more precisely, let's say:
L=0
f(x)=0 for all x element R, and therefore f'(x)=0 for all x element R
Now, from limit x->a_+ (f'(x) / g'(x)) = 0 , it should be possible to somehow prove, that there exists a delta>0, such that for all x element (a,a+delta), g'(x) is finite and non zero. I really don't see it though, why can g'(x) not be +oo somewhere in that interval?
Presuming:
(1), a is element of R (|a| =/= +oo)
(2), f and g are real functions
(3), limit x->a_+ (f'(x) / g'(x)) exists (must be element of R, or +-oo)
(4), limit x->a_+ (f(x)) = limit x->a_+ (g(x)) = 0
then
limit x->a_+ (f(x))/(g(x)) = limit x->a_+ (f'(x))/(g'(x))
I think I understand most of the proof but there's something right at the start that I'm completely stuck at and still don't understand precisely enough:
Let L=limit x->a_+ (f'(x) / g'(x)).
There exists delta>0, such that for all x element of (a,a+delta), f and g are both defined on this interval,
- I think this can be proved easily from (4), correct? Also, |f| and |g| are both smaller than some Epsilon>0. The following however, I don't understand at all:
and both f' and g' have a finite (not = oo or -oo) derivation on this interval, and also g'=/=0.
Why is the derivation necessarily finite?
EDIT:
To explain where I see the problem a bit more precisely, let's say:
L=0
f(x)=0 for all x element R, and therefore f'(x)=0 for all x element R
Now, from limit x->a_+ (f'(x) / g'(x)) = 0 , it should be possible to somehow prove, that there exists a delta>0, such that for all x element (a,a+delta), g'(x) is finite and non zero. I really don't see it though, why can g'(x) not be +oo somewhere in that interval?
Last edited: