difference between asymptotically large and polynomially large

Question

what is the difference between asymptotically large and polynomially large  how to find which function is asymptotically large or  polynomially large

Comments

Every polynomaily large is always asamptotiacly large but reverse is not true.

Answer 1

A function f(x) is polynomially larger than g(x) if it is greater only by some polynomial factor e.g. by n² , n³ but the exponent should be some constant not a function of n since then it will be even larger than factorial.

A function is asymptotically larger if it follows big -Oh notation .This is necessary and sufficient condition and here f(x) can be larger than g(x) by any factor , not necessarily polynomial.Let us understand what I have said through examples.

If f(x) =  n⁵ and g(x) =  n³logn , so removing common of n³ both sides , we have :

n²   which is polynomially larger than logn

So here f(x) is polynomially larger than g(x)

But if we take f(x)  =  nⁿ  and     g(x)  = n²

So now f(x) is not polynomially larger but we can say exponentially larger and to be generic asymptotically larger.

If we consider the other side , f(x) = nlogn    and   g(x) = n

Here also it is not so because the factor is logn which is less than polynomial but we can say asymptotically larger.

I hope you have understood the difference between the two.

Comments

so polynomial larger >  or < asmptotically larger

and in first paragraph what does   it will be even larger than factorial means here

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@Habib Asymptotically means comparing functions for large values of n. i.e., comparing two functions when $n \rightarrow \infty$ and polynomially means comparing two function between polynomial range i.e., In between polynomials. (Am i right ?)

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asymptotically larger "by polynomial factor" makes it polynomially larger.