Name for functions that are asymptotically equivalent to their shifted version

Question

I am working on limits and asymptotics and I would like to know (or where to look for) a correct definition of functions that are asymptotically equivalent (for $x\to \infty$) to shifted versions of themselves. That is, at least for sufficiently small $\delta$, $$\lim_{x\to \infty}\frac{f(x)}{f(x+\delta)} = 1,$$ such as $f(x) = x^r$ or $f(x) = \log^r(x)$. Is there a commonly accepted definition for this? Or is this strictly related to some other definition whom I am not aware of?

Answer 1

Note that the phrase "at least for sufficiently small $\delta$" isn't necessary. If the equation holds for some $\delta$, then $$ \lim_{x\rightarrow\infty}\frac{f(x)}{f(x+2\delta)}=\lim_{x\rightarrow\infty}\frac{f(x)f(x+\delta)}{f(x+\delta)f(x+2\delta)}\\ \qquad =\lim_{x\rightarrow\infty}\frac{f(x)}{f(x+\delta)}\cdot\lim_{x\rightarrow\infty}\frac{f(x+\delta)}{f(x+2\delta)}=1\cdot1=1, $$ so it holds for $2\delta$; and so on. So you can just say that $\lim_{x\rightarrow\infty}{f(x)/f(x+\delta)}=1$ for all $\delta$. And, as noted in a comment, this is is equivalent to saying that $$ \lim_{u\rightarrow\infty}\frac{f(\log u)}{f(\log u + \delta)}=\lim_{u\rightarrow\infty}\frac{f(\log u)}{f(\log (e^\delta u))}=1 $$ for all $\delta$, or for all positive $e^{\delta}$; which in turn is the same as saying that $L(u):= f(\log u)$ is a slowly varying function.