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.