Let all logarithms have base 2, and let all functions have domain the positive integers

Let all logarithms have base \(2\), and let all functions have domain the positive integers. For each pair \(f,g\) below, determine which of the relations \(f\in O(g)\), \(f\in\Omega(g)\), \(f\in\Theta(g)\), \(f\in o(g)\), and \(f\in\omega(g)\) hold. Prove every conclusion directly from the definitions of the five asymptotic relations. 1. \(f(n)=7n^2+3n+4\) and \(g(n)=n^2\). 2. \(f(n)=n^{3/2}\log n\) and \(g(n)=n^{3/2+1/100}\). 3. \(f(n)=2^{\sqrt{\log n}}\) and \(g(n)=(\log n)^{10}\). 4. \(f(n)=n\) for even \(n\) and \(f(n)=2n\) for odd \(n\), while \(g(n)=n\).

** Study \(f(n)/g(n)\), but translate each limit conclusion back into an eventual quantified inequality.