We will outline here the proof that the sequence {2ⁿ}_n=0,1,2,... does not converge, that is, it cannot have a finite limit. Indeed, think of any number L, say positive or zero. Pick any integer N larger than L, then 2^N is comfortably larger than L, so in particular it is larger than L + 1. If n > N, then 2ⁿ is even bigger, so the numbers 2ⁿ for n large are at least 1 away from L, thus this L cannot be a limit.

Since 2ⁿ > 0, these numbers will never get close to any negative L, so negative limits are also out of the question.