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.

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$f (x) = (x l o gx)^{2} - 4 \leq (x l o gx)^{2} + 4 (x l o gx)^{2} = 5 (x l o gx)^{2} \leq 5 x^{4}$

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

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

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