Suppose . Can you conclude from this that ?
I make this step in a somewhat longer proof in my homework, but I am not sure if this is true in general. If it is not true in general, what conditions do I need on the functions?
To me, everything seems consistent with the Properties of Limits, assuming you meant to have parenthesis around all of your functions.
ie.
Indeterminate Forms to understand what happens to these limits when dealing with dividing by 0 and infinity and such.
consider the following functions:
. then
, however:
, whereas .
one cannot apply the multiplicative/quotient properties of limits in this case, unless one is certain that the limits in the denominators are non-zero real numbers. in the example i have given, that is precisely the case for g(x) and h(x).
for another counter-example, suppose that:
.
again, we have:
, but
, whereas .
suppose that: .
to argue that L/M = NP/M implies L = NP, M cannot be 0, and must be finite.
and then to argue further that L = NP implies L/P = N, means P cannot be 0, or infinite. you can't divide by 0, or "cancel infinities".