Asymptotic Analysis of same-degree functions

Question

On a recent test, I was asked whether the following is true or false:

True or False: $10n^3 = O \left( 0.42n^3 \right)$

Comparing the two functions as n approaches infinity, I get:

$ \lim_{n \to \infty} \frac{10n^3}{0.42n^3} = \frac{10}{0.42} \lim_{n \to \infty} \frac{n^3}{n^3} = \frac{10}{0.42} \lim_{n \to \infty} 1 = \frac{10}{0.42} $

$\frac{10}{0.42}$ is a positive constant, so $10n^3 = \theta \left( 0.42n^3 \right)$

I put false--and was marked wrong. Am I wrong? If so, can you show me how?

Answer 1

Multiplication by a constant doesn't affect big-O fit. All polynomials of degree 3 are of the same order, so $10n^3 = O(0.42 n^3)$ is true.

Answer 2

@ConMan led me to the answer; I'll write it here just a little more directly:

$g(n) = \theta (f(n))$ implies both $g(n) = \Omega (f(n))$ and $g(n) = O (f(n))$ are true.

So, if we can show $g(n) = \theta (f(n))$ is true (which we have done), then both both $g(n) = \Omega (f(n))$ and $g(n) = O (f(n))$ are also true.