What is the value for the functions in the image when limit n tends to infinity?. Also what is the asymptotic complexity (big $O$ notation) for all the four functions?.
$$\begin{aligned}f_1(n) &= (n^{0.999999})\log n \\ f_2(n) &= 10000000 n\\ f_3(n) &= 1.000001^n\\ f_4(n) &= n^2\end{aligned}$$
It is readily apparent from the definition that \begin{align} f_1(n) &\in \mathcal O(n^{0.999999})\log n\\ f_2(n) &\in \mathcal O(n)\\ f_3(n) &\in \mathcal O(1.000001^n)\\ f_4(n) &\in \mathcal O(n^2). \end{align}
I will use $f_2$ as an example. Put $C = 10000000$ and $n_0=1$. Then for any integer $n\geqslant n_0$, $$f_2(n) = 10000000n \leqslant Cn,$$ so that $f_2(n)\in\mathcal O(n)$.