Question

Let `P_n=a^(P_(n-1))-1, AAn=2, 3, ...,` and let `P_1=a^x-1`, where a`inR^+`. Then what is the value of `lim_(x->0)P_n/x`?

Answer 1

`ln^n a`

Explanation:

`lim_(x->0)P_1/x = ln(a)`
`lim_(x->0)P_2/x = ln^2a`
`cdots`
`lim_(x->0)P_n/x = ln^n a`

NOTE

`lim_(h->0)(a^(0+h)-a^0)/h = ln(a)`