Is the Quotient test from shaums advanced calculus page 311-312 correct?
(a) If $f(x)\geq 0$ and $g(x)\geq 0$ for $a\leq x\leq b$, and if $\lim_{x\rightarrow a} \frac{f(x)}{g(x)}=A\neq 0$ or $\infty$ then $\int_a^b f(x)dx$ and $\int_a^b g(x)dx$ either diverge or converge.
(b) If $A=0$ in $(a)$ then $\int_a^b g(x)dx$ converges then $\int_a^b f(x)dx$ converges.
So if we let $g(x)=\frac{1}{x^2}$ and $f(x)=\frac{1}{x}$ then $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=0$ but $\int_0^1\frac{1}{x^2}dx$ and $\int_0^1\frac{1}{x}dx$ are divergent.
Well, there is certainly an unstated assumption in what you have posted that the only possible problem with the integrals converging is at $a$. For example, taking $f(x)=1/x$, $g(x)=1/x^2$ and $a=1,b=\infty$ then $f(x)/g(x)\to 1$ as $x\to a$.
But assuming that $\int_c^bf(x)dx$ and $\int_c^bg(x)dx$ exist and are finite for all $c\in (a,b)$, then (a) is correct. If $\lim_{x\to a}f(x)/g(x)=A\in(0,\infty)$ and $\int_a^bg(x)dx$ converges then for some interval $(a,c)$ we have $\int_a^cf(x)dx\leq \int_a^c2A g(x)dx$, which is finite.
There is a typo in (b), which should say if $\int_a^bg(x)dx$ converges then $\int_a^bf(x)dx$ converges. It says nothing about what happens if $\int_a^bg(x)dx$ diverges (as in your example).