Question

For any real number x, the sequence {(1+x/n)^(n)} as n approaches 1 to infinity. How did they get the answer of e^(x)? Please help me thank you :)))

Answer 1

\[\lim_{n \rightarrow \infty} (1+\frac{x}{n})^n=e^x\]
We can show this...
\[\lim_{n \rightarrow \infty}e^{\ln(1+\frac{x}{n})^n}=\lim_{n \rightarrow \infty}e^{n \ln(1+\frac{x}{n})}=\lim_{n \rightarrow \infty}e^\frac{\ln(1+\frac{x}{n})}{\frac{1}{n}}\]
use l'hosptial