## Calculus: Early Transcendentals (2nd Edition)

We will find the limit $\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}.$ 1) If it is equal to zero then $\ln x$ grows slower than $\ln x$; 2) If it is equal to $\infty$ then $\ln x^{20}$ grows faster than $\ln x$; 3) If it is equal to some ňon zero constant then their growth rates are comparable. We will use the logarithmic rule $\ln b^a=a\ln b$: $$\lim_{x\to\infty}\frac{\ln x^{20}}{\ln x}=\lim_{x\to\infty}\frac{20\ln x}{\ln x}=\lim_{x\to\infty}=20,$$ thus 3) is right and their growth rates are comparable.