Question 4.8.3: Examining the Limiting Behavior of ln x Use the properties ...

We can verify this as follows. First, recall that ln 3 ≈ 1.0986 > 1. Taking x=3^n, we have by the rules of logarithms that for any integer n,

\ln 3^n=n \ln 3.

Since x=3^n \rightarrow \infty, as n → ∞, it now follows that

\lim _{x \rightarrow \infty} \ln x=\lim _{n \rightarrow \infty} \ln 3^n=\lim _{n \rightarrow \infty}(n \ln 3)=+\infty,

where the first equality depends on the fact that ln x is a strictly increasing function.