We've got three different phrasings of L'Hôpital's rule and got a little bit confused about the subtle differences between them.
A
Let $-\infty \leq a < b \leq +\infty$ and let $f,g:]a,b[ \to \mathbb{R}$ be two functions, differentiable on $]a,b[$, and let g'(x) \not= 0 \forall x \in ]a,b[.
If further more on of the following cases applies
Case 1. $\lim\limits_{x \to a^+} f(x) = \lim\limits_{x \to a^+} g(x) = 0$.
Case 2. $\lim\limits_{x \to a^+} g(x) = \pm \infty$.
and L := \lim\limits_{x \to a^+} \frac{f'(x)}{g'(x)} exists,
then we have $\lim\limits_{x \to a^+} \frac{f(x)}{g(x)} = L$.
B
Let $-\infty \leq a < b \leq +\infty$ and let $f,g:]a,b[ \to \mathbb{R}$ be two functions, differentiable on $]a,b[$, and let g'(x) \not= 0 \forall x \in [a,b].
If further more on of the following cases applies
Case 1. $\lim\limits_{x \to a^+} f(x) = \lim\limits_{x \to a^+} g(x) = 0$.
Case 2. $\lim\limits_{x \to a^+} f(x) = \lim\limits_{x \to a^+} g(x) = \pm \infty$.
and L := \lim\limits_{x \to a^+} \frac{f'(x)}{g'(x)} exists,
then we have $\lim\limits_{x \to a^+} \frac{f(x)}{g(x)} = L$.
C
Let $a,b \in \mathbb{R}$ with $a < b$ and let $f,g:[a,b] \to \mathbb{R}$ be two functions, differentiable on $[a,b]$, and let g'(x) \not= 0 \forall x \in [a,b].
If further more $f(a) = g(a) = 0$ and L := \lim\limits_{x \to a^+} \frac{f'(x)}{g'(x)} exists,then we have $g(x) \not= 0 \forall x \in ]a,b]$ and $\lim\limits_{x \to a^+} \frac{f(x)}{g(x)} = L$.
A and B seem to be nearly the same
- A is lacking the condition for $\lim f(x)$ in case 2 - isn't this necessary?
- B requires g'(x) \not= 0 on the closed set $[a,b]$ - why is this? $g$ itself isn't even defined on $[a,b]$ and actually we're not working on the extended reals so the following leads to trouble g'(x) \not= 0 \forall x \in [-\infty,+\infty].
C is somehow different
- Since $a$ isn't allowed to be $-\infty$ and $b$ is not allowed to be $+\infty$, C is not including the same functions as A and B.
- It requires differentiability and g'(x) \not= 0 on the closed set $[a,b]$ in contrast to A and B.
- It lacks the second case. So C only helps us, if we can't calculate the limit of $\frac{f(x)}{g(x)}$ since $f(x)=0$ and $g(x)=0$.
- We additionally get information about $g(x)$ on the half-open interval $]a,b]$.
Maybe someone can help explaining the subtle differences and how the actually come into account.
Further more I noticed, that the rule is considering $\lim\limits_{x \to a+}$ - when applying this rule we often just write $\lim\limits_{x \to a}$. This is just being not completely strict?