infinity/infinity?

No. Infinity is not a number but an entity, which never ends. It has no value, and so infinity cannot be divided.

yes it is

There are a number of different types of infinity.  The first that is usually considered is that of the number of positive whole numbers.  This number is usually called aleph0 (or alephnull).  The mathematics of infinite numbers is weird, and is such that aleph0/aleph0 = aleph0.  

You cannot assume your intuition about, or knowledge of, finite numbers automatically applies to trans-finite numbers.

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