If as $x\to a$, $f(x)\to 0$ while $g(x)\to\infty$, then $\frac{f(x)}{g(x)}\to 0$. No L'Hospital's Rule needed, indeed L'Hospital's Rule could very well give the wrong answer, so must not be used.
The case "$\infty/0$" is more complicated. Again, L'Hospital's Rule is irrelevant, and must not be used. Suppose that as $x\to a$, $f(x)\to\infty$ and $g(x)\to 0$. If $g(x)$ approaches $0$ through positive values, one can conclude immediately that $\frac{f(x)}{g(x)}\to\infty$. If $g(x)$ approaches $0$ through negative values, one can similarly conclude that $\frac{f(x)}{g(x)}\to -\infty$. If, arbitrarily close to $x=a$, $g(x)$ can take on both positive and negative values, then the limit of $\frac{f(x)}{g(x)}$ does not exist.