Notation of One-Sided Limits

Question

If we had $\lim\limits_{x\,\rightarrow\,a^+} f(x)$, would the notation $\lim\limits_{x\,\downarrow\,a}$ be exactly the same as $\lim\limits_{x\,\rightarrow\,a^+}$, or is $\lim\limits_{x\,\rightarrow\,a^+}$ more general in that $x$ is nonincreasing when tending to $a$ whereas $\lim\limits_{x\,\downarrow\,a}$ implies that the $x$ must be decreasing when approaching $a$ from above?

Answer 1

The sequence formulation of $\lim\limits_{x\,\rightarrow\,a^+} f(x) = L$ is the following. For each sequence $\{x_n\},$ where $a < x_n$ for each $n$ and $x_n \rightarrow a,$ we have $\lim\limits_{n\,\rightarrow\,\infty} f(x_n) = L.$

The sequence formulation of $\lim\limits_{x\,\downarrow\,a} f(x) = L$ is the following. For each sequence $\{x_n\},$ where $a < x_{n+1} \leq x_n$ for each $n$ and $x_n \rightarrow a,$ we have $\lim\limits_{n\,\rightarrow\,\infty} f(x_n) = L.$

In the first case, we'll say that the unrestricted right limit of $f(x)$ as $x$ approaches $a$ is equal to $L.$ In the second case, we'll say that the monotone right limit of $f(x)$ as $x$ approaches $a$ is equal to $L.$ (These are terms I made up just now for use in this answer.)

I believe what you're asking is whether the unrestricted right limit notion is equivalent to the monotone right limit notion. I will show they are the same notion.

Assume the unrestricted right limit of $f(x)$ as $x$ approaches $a$ is equal to $L.$ Then the monotone right limit of $f(x)$ as $x$ approaches $a$ is also equal to $L,$ since every sequence $\{x_n\}$ having a monotone right-side approach to $a$ is a sequence having a right-side approach to $a.$

Assume the monotone right limit of $f(x)$ as $x$ approaches $a$ is equal to $L.$ To show that the unrestricted right limit of $f(x)$ as $x$ approaches $a$ is equal to $L,$ we need to show that whenever a sequence $\{x_n\}$ has a right-side approach to $a,$ then we have $\lim\limits_{n\,\rightarrow\,\infty} f(x_n) = L.$ For a later contradiction, assume that $\{x'_n\}$ is a sequence having a right-side approach to $a$ such that $\lim\limits_{n\,\rightarrow\,\infty} f(x'_n)$ is not equal to $L$ (the limit exists and is different from $L,$ or the limit doesn't exist). Passing to a subsequence if necessary, assume that $\lim\limits_{n\,\rightarrow\,\infty} f(x'_n) = M,$ where $L \neq M.$ [1] It now follows from a standard result that for every subsequence $\{x'_{n_k}\}$ of $\{x'_n\},$ we have $\lim\limits_{k\,\rightarrow\,\infty} f(x'_{n_k}) = M.$ But now, if we pick a subsequence that converges monotonically to $a$ (this can always be done), we get a contradiction, since we assumed that whenever a sequence has a monotone right-approach to $a,$ then the $f$-values of that sequence converge to $L$ (which is not equal to $M$).

[1] What I mean is that if the limit exists and is different from $L,$ then call this limit $M.$ And if the limit doesn't exist, then replace the sequence $\{x'_n\}$ with a subsequence that gives rise to a limit different from $L.$ This can always be done, for example, by using a subsequence that results in convergence to $\liminf\limits_{n\,\rightarrow\,\infty} f(x'_n)$ or by using a subsequence that results in convergence to $\limsup\limits_{n\,\rightarrow\,\infty} f(x'_n).$ At least one of these extreme limits has to be different from $L,$ otherwise the limit would exist and be equal to $L.$