 # Please explain how you attacked these questions step by step. 1) 1 is not O(1/x). 2) e^x is not O(x^5) Luz Stokes 2022-07-18 Answered
I'm studying for my discrete math class and I don't fully understand how to proof how a function is not a big O for certain questions. I understand that you have to assume that it is big O and proof by contradiction.
Please explain how you attacked these questions step by step.
1) 1 is not $O\left(\frac{1}{x}\right)$
2) ${e}^{x}$ is not $O\left({x}^{5}\right)$
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Step 1
1) Suppose $1=O\left(1/x\right)$. Then there exists a constant C such that for all large x, we have $1\le C/x$. However, for all $x>C$ we have $C/x<1$, a contradiction.
Step 2
2) Suppose ${e}^{x}=O\left({x}^{5}\right)$. Then there exists a constant C such that for all large x we have ${e}^{x}\le C{x}^{5}$. However, $\frac{{e}^{x}}{{x}^{5}}\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, a contradiction. [As Peter suggests, use L'Hopital's rule multiple times, or expand the Taylor series for ${e}^{x}$.]