Function that at $L_{2}$ but not in $L_{1}$ and also not in $L_{\infty}$

Question

as part of my effort to learn some functional analysis I'm trying to find an example to a function that is in $L^{2}$ $[0,\infty)$ but not in $L^{1}$ $[0,\infty)$ and also not in $L^{\infty}$ $[0,\infty)$.

I Thought of the following function: $\cfrac{1}{(t-1)}$ but this also not in $L_{2}$.

Would appreciate some help Thanks

Answer 1

Hint: if $f,g \geq 0$, $f \in L^{2}\setminus L^{1}$ and $g \in L^{2}\setminus L^{\infty}$ then $f+g$ would be such a function.