We commonly use the expression increases without bound to describe certain divergent behaviour of functions (e.g. the function $f(x)=x^2$ increases without bound on $[0,\infty)$). What would be the proper way of describing the behaviour of the curve of $f(x)$ if I want to move towards the right on $(-\infty,0]$?
In the curve sketching unit in my calculus course, I tend to describe all key elements (critical numbers intervals of increase/decrease, points of inflection, intercepts, etc.) from left to right on the $x$-axis, and so to increase consistency, I want to describe the increase/decrease of a function from left to right. So far, I have been saying "the function increases/decreases without bound towards the left", but this has been directly opposite to what that interval of the function is labelled. I want to know if there is better or more accurate language I could use.
To describe one-sided asymptotes, you can use one-sided limit notation. For example $$\lim_{x\to a^+}f(x)=\infty$$ is both brief and clear.
Or equivalently in words, as$\;x\;$approaches $a$ from the right, $f(x)$ approaches infinity.
Note that for the example above, we are describing right-to-left behavior, but that's the nature of the behavior.
To describe the behavior as $x$ approaches $\infty$ or $-\infty$, there is only one side to approach from, so there's no need to specify the direction of approach. For example, you might have $$\lim_{x\to -\infty}f(x)=\infty$$ or equivalently in words, as$\;x\;$approaches $-\infty$, $f(x)$ approaches $\infty$.
Note that for this example, once again we are describing right-to-left behavior.