## A challenging limit

*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:

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:

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

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

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