Let $(a,b)\subset \Bbb{R}$ be an interval and $f:(a,b)\to \Bbb{R}$ be a smooth function. Suppose that $f(x)>0$, $\forall x\in (a,b)$, and $\displaystyle\lim_{x\to b^-}f(x)=+\infty$. Let $t_0\in (a,b)$ be fixed. Is it possible that $$\int_{t_0}^bf(x)\,dx:=\lim_{t\to b^-}\int_{t_0}^{t}f(x)\,dx<+\infty?$$
Is there any example of this? Under which conditions on $f$ this happens or not?
Here is an example: $$ f(x) = |x|^{-1/2}, \qquad \lim_{x\to0^{-}} |x|^{-1/2}=+\infty, $$ while $$ \int_{-1}^0 |x|^{-1/2} \,dx = 2. $$
Under what conditions does this happen?
For the following particular family of functions, $$ f(x) = |x|^{-a}, \qquad a>0, $$ the integral is finite when $a<1$, while the integral does not exist $(\infty)$ when $a\ge 1$.