Finding the Limit at Infinity of \(f(x)=\ln x\)

Find \(\lim\limits_{x\rightarrow \infty }\ln x\).

Figure 60 \(f(x)=\ln x\)

Solution Table 17 and the graph of \(f( x) =\ln x\) in Figure 60 suggest that \(\ln x\) has an infinite limit at infinity. That is, \[\bbox[5px, border:1px solid black, #F9F7ED]{\lim\limits_{x\rightarrow \infty }\ln x =\infty} \]

TABLE 17
\(x\) \(e^{10}\) \(e^{100}\) \(e^{1000}\) \(e^{10{,}000}\) \(e^{100{,}000}\) \(\rightarrow x\) becomes unbounded
\(f( x) =\ln x\) 10 100 1000 10,000 100,000 \(\rightarrow f(x)\) becomes unbounded