Reading the Wikipedia article may help better understand the concept.

As to how we may define 1^∞, there's another reading there on how to define common operations on the set of real numbers extended with  ± ∞.

If you want function 1^x to be continuous in extended real number line you have to define 1^x for x =  + ∞ as and for x =  - ∞ as .

This reminds me of the exponential.

Limits of "1 power infinity" expressions often yield exponential functions. For example, for any given t:

Notice that the base approaches 1 and the exponent approaches infinity, but the resulting function is something completely different — and quite interesting.