I already know that we can compare the 'growth of the function' by little O or big O notation. The meaning of 'growth of the function' I mean like $\log x<x<e^x$ so $\textbf{basically how fast the function diverges}.$
I wonder if I can assign some algebraic object (hopefully real number) to such growth of the function so that we can 'quantitatively' compare how fast the function is by deciding one absolute value like a concept of 'velocity' in physics. Yes so basically I want to define a quantitative 'velocity' to the 'growth of the function'. So for example, $e^x$ diverges 'this' much faster than $x$.
Is this definition already exists? Or can we define such thing? Or is this meaningless?
