Clarification about notation for one-sided limits

Question

Is $\lim_{x \to 3-0} f(x)$ the same as $\lim_{x \to 3^-} f(x)$, and is $\lim_{x \to 3+0} f(x)$ the same as $\lim_{x \to 3^+} f(x)$?

Could anyone clarify this for me please? Thanks

Answer 1

$3-0=3+0=3$, so each of the corresponding limits are two sided limits. The other limits, $x\to 3^-$ and $x\to 3^+$, are one sided limits. In general, therefore, they are not the same.

Answer 2

The notations $\lim_{x \to a +0}$ and $\lim_{x \to a - 0}$ can be used to signify one-sided limits; the meaning is exactly the same as that of $\lim_{x \to a^+}$ and $\lim_{x \to a^-}$, respectively.

This is a so-called "abuse of notation"; one assigns to the string $a+0$ a meaning other than the usual one, "$a$ plus $0$."

The rational is to avoid a double-subscript and the assumption that no-one would normally write "$a+0$" there to mean the sum of $a$ and $0$, so that the intended meaning can be inferred from context.