Question

How do you describe the end behavior of `f(x)=x^10-x^9+5x^8`?

Answer 1

`f >=0`. As `x to +-oo, f to oo` See the graph.
The rate of rise of a function to `oo uarr`, in this order :
`ln x (x x^2 x^3 ...x^10...) e^x`

Explanation:

`f=x^8(x^2-x+5)=x^8((x-1/2)^2+19/4)>=0`

Also,

`f=x^10( 1-1/x+5/x^2 ) to oo`, as `x to +-oo`.

The rate of rise of a function to `oo uarr`, in this order :

ln x (x x^2 x^3 ,,,x^10...) e^x

As the derivatives of f, up to the order 10, are 0 at x = 0, the graph is

horizontal, at the 10-tuple point O. I would like to call this 10 as order

of `flatitude`.

Definition: If the derivatives of f are the same, up to and inclusive of order m at x = a, with `f^((m+1))(a)` different, the order of `flatitude` of the point x = a is m.

graph{x^8(x^2-x+5) [-5, 5, -2.5, 2.5]}