3.2 Asymptotics

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".

  1. \(f(x) = 2x^{3} + x^{2}\log x\)
    Exercise 1
    \(2x^{3}\) grows as fast as \(x^{3}\), and \(x^2\log x\) grows strictly slower than \(x^{3}\).

  2. \(f(x) =2x^{2} + x^{3}\log x\)
    Exercise 2
    \(2x^{2}\) grows as fast as \(x^{2}\), and \(x^{3}\log x\) grows strictly faster than \(x^{3}\) but strictly slower than \(x^{4}\).

  3. \(f(x) =(1.1)^{x}\)
    Exercise 3
    \((1.1)^{x}\) grows strictly faster than any polynomial.

  4. \(f(x) = (0.1)^{x}\)
    Exercise 4
    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).

  5. \(f(x) = \dfrac{x^{4} + x^{2} + 1}{x^{3} + 1}\)
    Exercise 5
    This fraction grows as fast as \(x^{4}/x^{3}=x\).

  6. \(f(x) = \dfrac{x^{4} + 5 \log x}{x^{4} + 1}\)
    Exercise 6
    This fraction grows as fast as \(x^{4}/x^{4}=1\).

  7. \(f(x) = 2^{3 \log_{2}x^{2}}\)
    Exercise 7
    \(2 ^{3 \log_{2}x^{2}} = 2^{\log_{2} (x^{2})^{3}} = 2^{\log_{2}x^{2 \cdot 3}} = 2^{\log_{2}x^{6}} = x^{6}\)