This post comes mostly from the youtube video by BlackPenRedPen found here: https://www.youtube.com/watch?v=89d5f8WUf1Y&t=3s

This in turn comes from Brilliant.com – details and links can be found in the original video

In this post we will have a look at a complicated-looking limit that has an interesting solution. Here it is:

\[\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}\]

This looks pretty daunting – but we will break the solution down into sections:

• taking the logarithms and rearranging
• recognising something familiar
• finding the numerical value

Step 1: Taking the Logarithm

The first step here is to take the logarithm, a generally useful trick when applying limits. First we assign the variable L to the limit (so that we can solve for it in the end). Now lets do some algebra:

\[L = \lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}}\]

\[\ln(L) = \ln(\lim_{n \rightarrow \infty} ( \frac{n!}{n^n})^{\frac{1}{n}})\]

Noting that the natural logarithm \(\ln\) is a continuous function and therefore we can take the limit outside of the function:

\[\ln(L) =  \lim_{n \rightarrow \infty} \ln( (\frac{n!}{n^n})^{\frac{1}{n}})\]

Next we can use the logarithm laws to bring down the exponent:

\[\ln(L) =  \lim_{n \rightarrow \infty}  \frac{1}{n} \ln(\frac{n!}{n^n})\]

Alright, now we have taken the logarithm, step 1 is complete.…