I'm trying to check if functions $f(n) = \Theta(g(n))$
- $f (n) = (4 \cdot n)^{150} + (2 \cdot n + 1024)^{400}$ vs. $g(n) = 20 \cdot n^{400} + (n + 1024)^{200}$
- $f (n) = n^{1.4} \cdot 4^n$ vs. $g(n) = n^{200} × 3.99^n$
- $f (n) = 2^{\log(n)}$ vs. $g(n) = n^{1024}$
In order to check the conditions, there have to be constants, $c_1, c_2$ and $n_0$ in which $c_1\cdot g(n) \leqslant f(n) \leqslant c_2 \cdot g(n)$ for all $n \geqslant n_0$.
Simply plugging in 1,2,3 for n in these functions after dividing f(n) by g(n) doesn't really help and as n approaches infinity and the functions 2 and 3' limits are 0, while function 1's limit is a very large number
How would I go about doing this?
Hint: if $$\lim_{n→\infty } \frac{f(n)}{g(n)} \ne 0,\infty$$ it automatically implies that $ f(n) = Θ(g(n))$.