When I reading through an article about asymptotic notation, there is a sentence - "For large enough inputs, the multiplicative constants and lower-order terms of an exact running time are dominated by the effects of the input size itself."
I get the part about "lower-order terms" since when lower order terms will become insignificant when compare with highest order term. But why "multiplicative constants" will be dominated? Why get rid off it will make sense?
I means no matter how big that X is, there will always be a noticeable difference between 3X and X.
This is due to the definition of asymptotic notations. If $f:\mathbb{N} \to [0,\infty[$, we have $$ \Theta(f) = \{g\mid \exists (c_0,c_1 > 0\mid c_0 f \leq g \leq c_1 f )\} $$ Clearly $\Theta (\lambda f) = \Theta (f)$ for any $\lambda > 0$, since we can scale $c_0$ and $c_1$ proportionally to $\lambda$.