Cardinals measure the number of elements in a set. Since there is no such thing as half an element of a set, there is absolutely no meaning to a reciprocal of a cardinal number. Certainly not an infinite cardinal.
Similarly, ordinal numbers measure the order type of an ordered set. Specifically, a well-ordered set. Since in this case there is no meaning to a reciprocal either, both of the equations $1-\frac1{2^{\aleph_0}}=1$ and $1-\frac1{2^\omega}$ are absolutely meaningless (also, note that cardinal and ordinal arithmetic are very different, $2^\omega=\omega$, whereas $2^{\aleph_0}>\aleph_0$).
It is tempting to say that in the surreal numbers one can find meaning to a reciprocal of an ordinal, but that would be false. Because the surreal numbers form a field, and as such their arithmetic is incompatible with the ordinal arithmetic, and with the cardinal arithmetic as well.
In short, there is no meaning to $\frac1{2^x}$ when $x$ is a cardinal or an ordinal, and there is even less meaning to $\lim_{x\to\infty}\frac1{2^x}$ when $x$ is either an ordinal or a cardinal, not because there is no meaning for limits. There is a meaning of limits in either context of cardinals and ordinals. But because (1) there is no meaning for reciprocals, and (2) there is no meaning for the $\infty$ symbol: do you mean the natural numbers, do you mean the entire ordinals/cardinals? The meaning of $\infty$ is unclear, and just today I was scolded by my colleague for using the $\infty$ symbol in a rather ambiguous set theoretic context.