The problem with accepting such notation as 1/ infinity, is that engenders the idea that infinity may be used within the structure of arithmetic. You then get such nonsense as infinity + 1.
Sure as a short hand it can be noted , however in rigourous calculus such notation is often more damaging than useful.
If we permit this kind of notation to enter mathematical proof sin(x)/x as x tends to infinity would be represneted as sin(infinity)/infinity. Then one might theorize that sin(infinity) is always less than 1 and so a finite number leading to the conclusion that the limit is 0. Where as the limit is obviously 1.
That same function sin(x)/x (xinR) is undefined and discontinuous at 0. Even tho the left and right limits as x tends to 0 are convergent and the same. It is a removeable singularity, The function needs to be extended to be continuous and defined on all real numbers.
The concept of Infinity needs to be carefully applied as more serious mistakes and errors in proof than that pointed out above have occured.
To wit(or not
)
1/infinity=0
so infinity/infinity=0*infinity. well 0 times anything = 0 and anything divided by anything =1. so 1 = 0.