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:
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At the conclusion of the statement of the theorem, Sohrab writes:" ... ... Note that, for finite a, we obviously have \(\displaystyle \lim{x \to a} = \lim{x \to a+}\) ... ... "I do not understand this remark.

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

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 \(\displaystyle \lim{x \to a} = \lim{x \to a+}\) ... ... "I do not understand this remark.

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

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 $\lim_{x\to -\infty+}$.