Section 5.5 Comparing Growth Rates
If,
\begin{equation*}
\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1
\end{equation*}
then \(f\) and \(g\) behave in essentially the same way “at \(\infty\)”. For example, if \(g\) has a limit at infinity, i.e. \(\lim_{x \to \infty} g(x) = L\text{,}\) then,
\begin{align*}
\lim_{x \to \infty} f(x) \amp = \lim_{x \to \infty} \underbrace{\frac{f(x)}{g(x)}}_{\to 1} \cdot \underbrace{g(x)}_{\to L}\\
\amp = 1 \cdot L\\
\amp = L
\end{align*}
