Question

3.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

Definition: Let $f$ and $g$ be real-valued functions. We say that $f(x)$ is $O(g(x))$ is there exists two numbers $C$ and $k$ such that $|f(x)| ≤ C|g(x)| \forall x > k.$ or in limit terms $\lim\limits_{x\to+\infty}|\frac{f(x)}{g(x)}|<C.$

Let's consider $\lim\limits_{x\to+\infty}\frac{(xlogx)^2−4}{x^4}=\lim\limits_{x\to+\infty}\frac{log^2x}{x^2}-\lim\limits_{x\to+\infty}\frac{4}{x^4}=0$ . So $f(x)=(xlogx)^2−4$ is $o(x^4).$

Let's consider $\lim\limits_{x\to+\infty}\frac{5x^5}{x^4}=\lim\limits_{x\to+\infty}5x=+\infty.$ So for all C $\exist k |5x^5|>C |x^4|$ for all $x>k$ . So, $g(x)=5x^5$ is not an $O(x^4)$ .