Question

evaluate limit of n->infinity of (1)/[(n-4)/(n)]^n

Answer 1

\[\lim_{n\rightarrow\infty}\frac{1}{\left(\frac{n-4}{n}\right)^n}=\lim_{n\rightarrow\infty}\left(\frac{n}{n-4}\right)^n=\lim_{n\rightarrow\infty}\left(1+\frac{4}{n-4}\right)^n=\]\[=\lim_{n\rightarrow\infty}\left[\left(1+\frac{4}{n-4}\right)^\frac{n-4}{4}\right]^\frac{4n}{n-4}=\exp{\left(\lim_{n\rightarrow\infty}\frac{4n}{n-4}\right)}=e^4\]