What's this equal: $\lim_{p\to\infty} \big(1+\frac{1}{||f||_p})^{||f||_p}?$

Question

let $a , b$ two real numbers such that $a < b$ and $f\in L^\infty([a,b])$. I want to evaluate this limit:

$$\lim_{p\to\infty} \left(1+\frac{1}{||f||_p}\right)^{||f||_p}.$$

I know that $\lim_{p\to+\infty}\|f\|_p=\|f\|_\infty,$ which is finite, but I can't predict exactly what this limit is.

Answer 1

Since the interval is finite and $f$ itself is in $L^\infty$, all the norms in question are finite so you just get $(1+1/\| f \|_\infty)^{\| f \|_\infty}$.