Infinities, Large and Small

     Infinities are enigmas that hide in the formulae of theoretical physicists waiting to pop out near the end of lengthy calculations making their efforts meaningless. Infinities stymie our investigative efforts and poison our logic yet we  give them status by assigning them their own mathematical symbol and by giving them names like singularities and black holes to make their embarrassing snubs appear to be logical conclusions.
     My keyboard has no infinity symbol but we all know that it looks like a twisted circle creating an intersection of lines at its center. The symbol was supposedly chosen to point out the fact that when one reaches the end they are back at the beginning, an appropriate reprimand for mathematicians and physicists that have followed their symbols to a dead end.  The pursuits leading to infinities seems to have become a game played by theorists that is lost whenever one wins.
     There are many functions that lead to infinities. Some take a diminishing path to reach the absurdity while others take an expansive path. Are there, then, large and small infinities?

       comments welcomed, especially in mathematical form