## Practice with Big O

Find the least nonnegative integer, \(n\), such that \(f(x\)) is \(O(x^{n})\) when \(f\) is defined by each of the expressions below.

*If there is none, enter "none".*

- \(f(x) = 2x^{3} + x^{2}\log x\) \(2x^{3}\) grows as fast as \(x^{3}\), and \(x^2\log x\) grows strictly slower than \(x^{3}\).
- \(f(x) =2x^{2} + x^{3}\log x\) \(2x^{2}\) grows as fast as \(x^{2}\), and \(x^{3}\log x\) grows strictly faster than \(x^{3}\) but strictly slower than \(x^{4}\).
- \(f(x) =(1.1)^{x}\) \((1.1)^{x}\) grows strictly faster than any polynomial.
- \(f(x) = (0.1)^{x}\) As \(x\) goes to infinity, \((0.1)^{x}\) goes to 0. So it grows strictly slower than any constant (same as a polynomial of degree 0).
- \(f(x) = \dfrac{x^{4} + x^{2} + 1}{x^{3} + 1}\) This fraction grows as fast as \(x^{4}/x^{3}=x\).
- \(f(x) = \dfrac{x^{4} + 5 \log x}{x^{4} + 1}\) This fraction grows as fast as \(x^{4}/x^{4}=1\).
- \(f(x) = 2^{3 \log_{2}x^{2}}\) \(2 ^{3 \log_{2}x^{2}} = 2^{\log_{2} (x^{2})^{3}} = 2^{\log_{2}x^{2 \cdot 3}} = 2^{\log_{2}x^{6}} = x^{6}\)