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.