# “Small” real numbers [duplicate]

At least for me, my intuition for what numbers are large or small comes entirely from positive numbers.

I find it challenging to use the word "small" correctly when talking about negative numbers. Relative to $-1$, is $-1000$ small or is $-0.001$ small?

Interpretation (1): Smallness is defined by the less than relation: $x$ is smaller than $y$ iff $x < y$. In this interpretation, $-1000$ is a small number, while $-0.001$ is not.

Interpretation (2): Smallness is defined by proximity to $0$: $x$ is smaller than $y$ iff $|x| < |y|$. In this interpretation, $-0.001$ is a small number, while $-1000$ is not.

I find myself often getting tongue tied when I try to describe computations of $\lim\limits_{x \rightarrow -\infty} f(x)$ or $\lim\limits_{x \rightarrow 0^-} f(x)$. How do you handle talking about such things? Do you adopt one of the above interpretations? Or some other approach? I am thinking in particular of freshman calculus classes in the United States, where limits are typically presented at an intuitive, but not rigorous, level.