Use the definition of f(x) is O(g(x)) to show that x^4+9x^3+4x+7 is O(x^4).

What we want here is to use estimates rather than the theorem proved in class/book.

By the definition we must find a C and k so that for all x> k
x^4+9x^3+4x+7< C*x^4

The proof of the theorem of the book suggests that we should just take C to be 1+9+4+7=21 and then k=1 (i.e. the sum of the absolute values of the coefficients). This certainly works and is the simplest, but for the sake of learning a little more (we'll almost show, in addition, that x^4 is not O( 9x^3 +4x+7)).

Let us take C=2 and compute a k so that
x^4+9x^3+4x+7< 2*x^4 for all x> k.

This is equivalent to (subtract x^4 from both sides) asking that
9x^3+4x+7< x^4 for all x> k. Well (this is not the smallest k) just take k at least as large as 3*9=27. Then, for x> k, we have x^4 > x*x^3> 27*x^3 > 9x^3 + 9x^3+9x^3 > 9x^3 + 4*x +7 as required.