Behavior of a function near a singular point

Question

What is the correct wording concerning the behavior of a function near a singular point? For example, the function $$f(x)=\frac{e^x+2}{x^2}$$ behaves like $2/x^2$ as $x$ approaches zero. Often, one is only interested at how fast it "explodes". So is it acceptable to say that "$f$ behaves like $1/x^2$ as $x$ approaches zero"? Or perhaps "$f \sim 1/x^2$ as $x\rightarrow 0$"? I would like to "ditch" the constant because, in the manuscript I am writing, it makes the text unnecessarily complicated and the constant of proportionality has no importance. At least in my case.

Answer 1

We can use both

• asymptotic notation $f(x)\sim \frac1{x^2}$

• big O notation $f(x)= O\left(\frac1{x^2}\right)$

Answer 2

You may want to use Landau notation: $$f\in\mathcal O\left(\frac1{x^2}\right) \text{ as } x\to 0.$$