MHB L'Hopital's Rule _ Statement of Theorem (Houshang H. Sohrab)

[Math Amateur]

I am reading Houshang H. Sohrab's book: Basic Real Analysis (Second Edition).

I need help with an aspect of Sohrab's statement of Theorem 6.5.1 (L'Hopital's Rule) on pages 262-263. Sohrab's statement of Theorem 6.5.1 reads as follows:
View attachment 3935
https://www.physicsforums.com/attachments/3936
At the conclusion of the statement of the theorem, Sohrab writes:" ... ... Note that, for finite a, we obviously have $$\lim{x \to a} = \lim{x \to a+}$$ ... ... "I do not understand this remark.

Surely since $$f, g$$ are defined on $$(a, b)$$ the whole statement of the Theorem should be in terms of limits of the form $$\lim{x \to a+}$$ ... indeed for a function defined on $$(a,b)$$ it does not seem right to me to talk about limits of the form $$ \lim{x \to a}$$?

Can someone please clarify this issue for me?

Peter

 

[caffeinemachine]

I am reading Houshang H. Sohrab's book: Basic Real Analysis (Second Edition).

I need help with an aspect of Sohrab's statement of Theorem 6.5.1 (L'Hopital's Rule) on pages 262-263. Sohrab's statement of Theorem 6.5.1 reads as follows:At the conclusion of the statement of the theorem, Sohrab writes:" ... ... Note that, for finite a, we obviously have $$\lim{x \to a} = \lim{x \to a+}$$ ... ... "I do not understand this remark.

Surely since $$f, g$$ are defined on $$(a, b)$$ the whole statement of the Theorem should be in terms of limits of the form $$\lim{x \to a+}$$ ... indeed for a function defined on $$(a,b)$$ it does not seem right to me to talk about limits of the form $$ \lim{x \to a}$$?

Can someone please clarify this issue for me?

Peter

I think it is perfectly alright to write $\lim_{x\to a}$ even though the function is not defined on the left of $a$ (and on $a$).
It will just be intepretted as $\lim_{x\to a^+}$.
The remark by Sohrab is there, I guess, because it doesn't mean anything to write $\lim_{x\to -\infty+}$. So when $a=-\infty$, $\lim_{x\to a}$ has to be interpretted as $\lim_{x\to -\infty}$ and not, of course, as $\lim_{x\to -\infty+}$.