Showing a Sequence Converges

Show \(\left\{\ln \left(\dfrac{2}{n}+3\right) \right\}\) converges and find its limit.

Solution Since \(\lim\limits_{n\rightarrow \infty }\left( \dfrac{2}{n}+3\right) =3\) [from Example 5(a)], the sequence \(\{ s_{n}\}=\left\{ \dfrac{2}{n}+3\right\} \) converges to \(3.\) The function \(f(x) =\ln x\) is continuous on its domain, so it is continuous at \(3\). Then \[ \lim\limits_{n\rightarrow \infty }f( s_{n}) =\lim\limits_{n\rightarrow \infty }f \left( \dfrac{2}{n}+3\right) =\lim\limits_{n\rightarrow \infty }\ln \left( \dfrac{2}{n}+3\right) =\ln\left[\lim\limits_{n\rightarrow \infty } \left(\dfrac{2}{n}+3\right) \right] =\ln 3 \]

So, the sequence \(\left\{\ln \left( \dfrac{2}{n}+3\right) \right\}\) converges to \(\ln 3\).