How to solve this big O notation of a fraction I want to estimate with big O the following expressions:

$$
\begin{align*}
f_1(x) &amp;= \frac{x^4 + x^2 + 1}{x^4 + 1}, \\
f_2(x) &amp;= \frac{x^3 + 5 \log x}{x^4 + 1}.
\end{align*}
$$

How do you eliminate the constant on the bottom of the fraction? How would you work to solve these types of problems?
The idea is that $x^4 + x^2 + 1 = \Theta(x^4)$, and similarly $x^4 + 1 = \Theta(x^4)$. So the first expression is $\Theta(x^4)/\Theta(x^4) = \Theta(1)$. The second expression can be estimated similarly.

This sort of calculation shows the advantage of asymptotic notation - irrelevant details disappear and calculations become manageable.