Notation and semantic for approximation?

Question

Likewise the notation $\approx$ that says is "approximately equal to.." is there a symbol for the meaning "is approximately less of..."? What's the formal meaning of such symbols?

Answer 1

I don't know how to do it in LaTeX, but yes - just replace the underbar in $\leq$ with a squiggle and that's it. It is used when you have bounds, but the bound is more conveniently expressed as an approximation. We used it when we did numerical error analysis using Taylor approximations, and also when something is bound by a random variable that is approximated by a normal or something.

Edit: found the LaTeX command, it's '\lesssim':

$f(x)\lesssim C \Leftrightarrow f(x) \leq C \pm \epsilon$, where $\epsilon$ is comparatively small and might even vanish asymptotically.

Answer 2

$$\lesssim\text{or} \gtrsim $$