Question

Using the definition of "Big-O" determine if each of the following functions, f(x) = (xlogx)^2 - 4 and g(x) = 5x^5 are O(x^4) and prove your claims.

Answer 1

$f(x) = (xlogx)^2 - 4\le (xlogx)^2 +4 (xlogx)^2 =5 (xlogx)^2 \le5x^4$

$f(x)=O(x^4)$

$\displaystyle{\lim_{x\to \infin}}(g(x)/x^4)=\displaystyle{\lim_{x\to \infin}}(5x^5/x^4)=\infin$

$g(x)\neq O(x^4)$