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 $\ln 3$.