Answer
$$A = P{e^{rt}}$$
Work Step by Step
$$\eqalign{
& A = P{\left( {1 + \frac{r}{n}} \right)^{nt}} \cr
& \mathop {\lim }\limits_{n \to \infty } A = \mathop {\lim }\limits_{n \to \infty } P{\left( {1 + \frac{r}{n}} \right)^{nt}}{\text{ }} \cr
& = P\mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{r}{n}} \right)^{nt}} \cr
& {\text{Evaluating the limit}} \cr
& \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{r}{n}} \right)^{nt}} = {\left( {1 + \frac{r}{\infty }} \right)^\infty } = {1^\infty } \cr
& {\text{This limit has the form }}{1^\infty }{\text{ }} \cr
& {\left( {1 + \frac{r}{n}} \right)^{nt}} = {e^{nt\ln \left( {1 + \frac{r}{n}} \right)}},{\text{ then}} \cr
& \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{r}{n}} \right)^{nt}} = \mathop {\lim }\limits_{n \to \infty } {e^{nt\ln \left( {1 + \frac{r}{n}} \right)}} = {e^{\mathop {\lim }\limits_{n \to \infty } nt\ln \left( {1 + \frac{r}{n}} \right)}} \cr
& {\text{The first step is to evaluate }} \cr
& L = \mathop {\lim }\limits_{n \to \infty } nt\ln \left( {1 + \frac{r}{n}} \right) = t\mathop {\lim }\limits_{n \to \infty } \frac{{\ln \left( {1 + \frac{r}{n}} \right)}}{{1/n}} = t\left( {\frac{{\ln \left( {1 + \frac{r}{\infty }} \right)}}{{1/\infty }}} \right) = \frac{0}{0} \cr
& {\text{Using L'Hopital's rule}} \cr
& L = t\mathop {\lim }\limits_{n \to \infty } \frac{{\frac{d}{{dn}}\left[ {\ln \left( {1 + \frac{r}{n}} \right)} \right]}}{{\frac{d}{{dn}}\left[ {1/n} \right]}} = t\mathop {\lim }\limits_{n \to \infty } \frac{{\frac{{ - r/{n^2}}}{{1 + 1/n}}}}{{\left( { - 1/{n^2}} \right)}} \cr
& = t\mathop {\lim }\limits_{n \to \infty } \frac{r}{{1 + 1/n}} \cr
& = t\left( {\frac{1}{{1 + 1/\infty }}} \right) = rt \cr
& {\text{Therefore,}} \cr
& \mathop {\lim }\limits_{n \to \infty } {e^{nt\ln \left( {1 + \frac{r}{n}} \right)}} = {e^{\mathop {\lim }\limits_{n \to \infty } nt\ln \left( {1 + \frac{r}{n}} \right)}} = {e^{rt}} \cr
& \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{r}{n}} \right)^{nt}} = {e^{rt}} \cr
& and \cr
& A = P\mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{r}{n}} \right)^{nt}}{\text{ }} = P{e^{rt}} \cr
& A = P{e^{rt}} \cr} $$
