- When I read the proof of L'Hospital's Rule, the exact meaning of "x tends to a" made me confused.
- I'll write briefly about the proof:
$\lim_{x \to a^+}f(x) = \lim_{x \to a^+}g(x) = 0$ and $f(a) = g(a) = 0$ according to Cauchy Theorem $ \exists c \in (a,x) , \frac{f(x)}{g(x)}= \frac{f '(c)}{g'(c)} $
As x tends to a from the right, the value c also tends to a from the right, and the desired conclusion follows:
$ \lim_{x\to a^+} \frac{f(x)}{g(x)}=\lim_{x\to a^+} \frac{f'(c)}{g'(c)}=\lim_{c\to a^+} \frac{f'(c)}{g'(c)}=L$
- My first question: I can feel "let x tends to a then c tends to a" or "let $x \to a^+,then\ c \to a^+$" means when x get closer to a, then c close to a.What is the exact meaning of this?Can this translate into exact math languge?I don't think that's very precise.
- My second question: $\forall x>a, \exists c \in (a,x), \frac{f(x)}{g(x)}= \frac{f '(c)}{g'(c)} $ Using this property, how to get "$\lim_{x\to a^+} \frac{f(x)}{g(x)}=\lim_{x\to a^+} \frac{f'(c)}{g'(c)}=\lim_{c\to a^+} \frac{f'(c)}{g'(c)}$",just let x tends to a from the right? I think there are some steps that are missing.