The first set of symbols you posted is read as "the limit as x increases without bound of a/x." In technical terms, it is defined as the number L such that for every positive difference e, there is some number N such that |L - a/x| < e for every x > N, if any such number L exists.
Graphically, this is a horizontal asymptote, and in the function you gave, the horizontal asymptote is 0.
The second symbol you posted is not defined as an operation in the real number system as is not a real number. There are infinite numbers in other number systems, but you lose some of the algebraic structure of the real numbers, making some statements much more convoluted or simply not true.
a/0, which is defined to be is not a real number, because 0 has no mutiplicative inverse. that is, there is no real number such that .
In limiting behavior, the limiting form a/0 is not indeterminate, it signifies that no limit exists. Note that when you write "infinite" limits, you are writing that the limit does not exist in a particular way. A limit is always a real number.
In this case, consider . From the left, this limit increases without bound, while from the right this limit decreases without bound. Thus there is no limit or defined limiting behavior at 0, even though it has the form 1/0.