A century is quite a lot of time really
We can't say with certainty whether Bitcoin or the US dollar (or even the US as such) will still exist in so much time. Wtf, we can't even say that either will still be around in just a decade (the latter more concerns Bitcoin, of course). Note that Bitcoin itself doesn't need to compete out the dollar since the latter can die on its own or just become a second grade currency like Venezuelan bolivar (the Zimbabwean dollar being the extreme but this is a big distance). The same can well happen to Bitcoin too, obviously
This is not so over long enough timeframe
We are just lucky to live in times when great technology advances are made but they are all based on practical application of fundamental discoveries made earlier (say, some 50 or even 100 years ago). But these discoveries don't in any way guarantee that they can and will be used in real life, or there is much left to discover at all in the future. For example, the nuclear fusion has been known maybe for 100 years already, but we still didn't advance past using it only in thermonuclear weapons, and it may take another 100 years till we can use nuclear fusion for producing energy. In other words, no one can say whether the technology will move ahead as fast as it has been moving for the last 20 years