Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$

Question

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$.

I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.

Answer 1

Use the dominated convergence theorem with $\max(1,|f(x)|^2)$ as the dominating function.