Suppose $f:(a,b] \to \mathbb R$ is a continuous function and $\lim_{x\to a+} f(x)=+\infty$
Suppose $f(x)\leq g(x)$ for all $x\in(a,b]$, for some $g：[a,b]\to \mathbb R$ satisfying $\int_a^b g(x)dx<+\infty$
Can we conclude that
$\lim_{t\to a^+}\int_t^b f(x)dx \leq \int_a^b g(x)dx$ ?
Thanks.
Yes, we can. First, for sufficiently small $h>0$, we have $$\int_{a+h}^bf(x)\,dx\leq\int_{a+h}^bg(x)\,dx$$ and since the r.h.s tends to a limit as $h\to 0$, we conclude that the l.h.s is bounded. Moreover, since $f(x)\to\infty$ as $x\to a^+$, we can choose $h>0$ so that $f(x)$ is positive in $(a,a+h)$, and in particular, the integral $\int_{a+h}^bf(x)\,dx$ increases as $h$ decreases to zero. Thus, the function $h\to\int_{a+h}^bf(x)\,dx$ is monotonically increasing and bounded as $h\to 0^{+}$, hence the limit exists, and obviously is not larger than the integral of $g$.