A minor point- but exponential curves never actually asymptote or go vertical, they just keep getting closer to vertical forever. But with a big enough level of growth it might as well be vertical.
True. Making any exponential curve look like a hockey stick is mostly a matter of picking the scale. I could probably create a chart showing how the population of the earth went vertical during the reign of the Caesars. Of course, a logarithmic plot solves that issue.