In theory, graphs such as the inverse exponential function never become parallel to the x-axis, and instead become arbitrarily close to being parallel.
In practice, because there are only 53 bits of numeric precision in floating point numbers, when the relative change in slope becomes less than 2^(-53) then it would become numerically parallel. However, people seldom plot graphs out that far.
There is another limit in practice, which is that at some point such a plot would maintain the same plotted vertical pixel difference for the remainder of the graph. Is that the point you are looking for? If so then it will depend upon the graphics resolution you are using and upon the aspect ratio of the area you are plotting into, so resizing the plot would change the answer, which would not seem to be a robust measure.
