Proof of L'Hospital's Rule 3: Lim f(x)/g(x) = ∞

[ocalvino]

if lim f(x)= infinity= lim g(x)
x->infinity x->infinty

and lim f'(x)/g'(x)=infinity
x-> infinity

then lim f(x)/g(x)=inifity
x-> inifinity

The above fact is what I am trying to prove. From my notes, i see the following:

For m>0, choose k>0, such that if x> k* and g(x),f(x)>0,

then f'(x)/g'(x)> m(4/3).

this is actually where i get lost (so early into the process). can someone explain to me where exactly the prof is headed to with this info? also, is k a functional value through m? if so...how do i choose such k?

 

[ocalvino]

i just realized that my post is quite confusing. i suppose I am not too sure what to ask. If you don't understand my original post, perhaps u can just help me start off the proof. thanks.

 

[ocalvino]

i also have the following hints to use after the hints in the original post:

fix a>k, then by cauchy mvt,

f(x)-f(a) f'(c)
_______ = ____ > m(4/3) for x>a>k*, and such that
g(x)-g(a) g'(c)

g(x)>g(a)>0 and f(x)> f(a)> 0 where c>a>k*