conceptual question reguarding limits

What is the difference between $\displaystyle lim_{x\to\infty}\frac{a}{x}$ and $\displaystyle \frac{x}{\infty}$? When I first learned limits they were defined to me as when the independent variable of the function grows arbitrarily close to the value of the limit. Infinity was defined to me as when a number grows arbitrarily large. With this in mind, it makes very little sense to talk about a limit going to infinity.

On a similar note, why is $\displaystyle \frac{a}{0}$ indeterminant? Wouldn't it make more sense if it were $\displaystyle \infty$?