I often read $f(x) \sim g(x)$ and I wonder what the Standard Interpretation of this $\sim$ is. It seems to mean something like asymptotically equally distributed, something like $f(x)=g(x)(1+o(1))$.
But I wonder, if $f(x)= c g(x)$, is it true that then $f(x) \sim g(x)$? Because with the above Definition with $o(1)$ this would not be the case?
I ask because of the following example: It is written: $${{k-1} \choose {x-1}} p^{x} (1-p)^{k-x} \sim {{k-1} \choose {x-1}} p^x$$
Now I wonder whether it holds that $$ c {{k-1} \choose {x-1}} p^{x} (1-p)^{k-x} \sim {{k-1} \choose {x-1}} p^x$$
Near x=a : $ f(x) \sim g(x) $ means $ \lim_{x \rightarrow a} \frac{f(x)}{g(x)} = 1 $
See: http://en.wikipedia.org/wiki/Asymptotic_analysis