EXPF04 In the previous problem what would be the balance at the end of the 5 years if the 6% compounding were increased to instantaneous compounding?

This looks so much like the definition of e, define k/r as n  so the expression for the amount reads

Now, knowing the laws of exponents, A can be written as 

As the number of intervals increases (k increases and k/r increases) the compunding approaches instantaneous and the expression in the brackets becomes 

So in the limiting case of instantaneous compounding the amount is 

For this problem with P = $1000, r = 0.06, and t = 5 years, the maximum balance for instantaneous compounding is