let $ f(x)=\frac{1}{1+x^{2}}$ find $\lim _{p \to \infty } \|f \|_{p}$?

Question

Consider $[0,1]$ with Lebesgue measure, and let $f:[0,1] \rightarrow \mathbb{R}$ , $f(x)=\frac{1}{1+x^{2}}$. How do we find $\lim _{p \to \infty } \|f \|_{p}$ ?

1) $0$

2) $\frac{\pi}{2} $

3) $1$

4) $\infty$

$\|f\|_p=\left(\displaystyle\int|f|^{p}d\mu\right)^{1/p}$ for $p=1$ we have

$\displaystyle\int \frac{1}{1+x^{2}}\, d x=\tan ^{-1}(x)+C$

Answer 1

This function is both continuous and positive on $[0,1]$. Hence, $$\lim_{p\rightarrow\infty}\|f\|_p=\|f\|_\infty:=\max_{x\in [0,1]} |f(x)|.$$ Can you compute the right-hand side?

Hopefully, you can see rather easily from this that the answer must be

$3)\ 1$