TITLE: Help solving the limit $\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n}$
QUESTION [8 upvotes]: $$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} $$
I've tried:
$$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} = \lim_{n\to \infty} e^{\ln{\left(1+\frac{1}{n!}\right)^{2n}}} = \lim_{n\to \infty} e^{2n\ \ln{\left(1+\frac{1}{n!}\right)}}$$
But I don't know how to work with the factorial

REPLY [2 votes]: Here, I complete @milo's try: (we use $\ln(1+x)\sim_0x)
$$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} = \lim_{n\to \infty} e^{\ln{\left(1+\frac{1}{n!}\right)^{2n}}} = \lim_{n\to \infty} e^{2n\ \ln{\left(1+\frac{1}{n!}\right)}}=\lim_{n\to \infty} e^{\frac{2n}{n!}}=e^0=1.$$