Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Solution:

$\dfrac{x^5+10x^3+x+1}{x^4}= \dfrac{x^5}{x^4}+\dfrac{10x^3}{x^4}+\dfrac{x}{x^4}+\dfrac{1}{x^4}=x+\dfrac{10}{x}+\dfrac{1}{x^3}+\dfrac{1}{x^4}$

$\implies x+10x^{-1}+x^{-3}+x^{-4}$

Degree of polynomial = 1

So, it is not $O(x^4)$