3. Application of Asymptotes and Growth Rates

Growth Rates

Definition

Consider functions $p_A(x)$ and $p_B(x)$ that grow as $x$ increases.

If

${\lim }_{x\to \infty }\frac{p_A(x)}{p_B(x)}=0$

then $p_B(x)$ has a higher growth rate than $p_A(x)$.

If

${\lim }_{x\to \infty }\frac{p_A(x)}{p_B(x)}=c>0$ then $p_A(x)$ has the same growth rate as $p_B(x)$.

If $\frac{p_A(x)}{p_B(x)}$ keeps increasing then $p_A(x)$ has a higher growth rate than $p_B(x)$.