Notation for asymptotic approximation

Question

I was reading Stirling's approximation and got quite confused with the idea of asymptotic formula. So in Wikipedia it says that a function $F(n)$ of $n$ is asymptotic formula for $P(n)$ if $P(n)$ is asymptotically equivalent to $F(n)$ that is if $$\lim_{n\rightarrow\infty}\frac{P(n)}{F(n)}=1$$ my question is what is difference between this and $$F(n)\rightarrow P(n)$$

Answer 1

The limit of a quotient of functions is the quotient of the limits (if the limits exist) so both way of writing would mean the same.

The problem of the second expression is that

$f(n) \rightarrow P(n)$ when $n \rightarrow \infty$ has no sense since the right part should be a number! (or you are speaking of convergence and f(n) is more a $f_n$. Then you have to take care of which convergence you want to deal with).

I hope i'm clear: don't use the latter ;)

Btw, @Raymond Manzoni 's answer is also a good one but i wanted to make clearer the non-sense of the latter expression

Answer 2

Consider $F(n) = n$ and $P(n) = n+\sqrt{n}$.

Note that $$\lim_{n \to \infty} (P(n)-F(n)) = +\infty \implies F(n) \not\to P(n)$$ whereas $$\lim_{n \to \infty} \dfrac{P(n)}{F(n)} = 1 \implies F(n) \sim P(n)$$

As $n \to \infty$, the statement that $F(n) \to P(n)$ is a more stronger statement than $\displaystyle\lim_{n \to \infty} \dfrac{P(n)}{F(n)} = 1$.

Answer 3

Note that we can reformulate $$\lim_{n\rightarrow \infty}\frac{P(n)}{F(n)}=1,$$ with a little bit of algebra as $$P(n)\underset{n\rightarrow\infty}{\sim} F(n)\Leftrightarrow \lim_{n\rightarrow \infty}\frac{P(n)-F(n)}{F(n)}=0.$$

That is the difference between the functions becomes negligible as $n\rightarrow\infty$.

For example consider $F(n)=10^n$ and $P(n)=10^n+1$. As $n$ gets big, the difference between them becomes negligible.

Similarly if you have €1,000.42 in your bank account you would just say you have €1,000 as the difference between them is relatively negligible.