ff = ((1 + b)^2)/Sqrt[1 - b*b]

(1 + b)^2/Sqrt[1 - b^2]

LogLogPlot[ff, {b, .9999999999999999999, .99999999999999999999}]

The limits of the plot are arbitrary here but need to be many nines larger for the circumstances I am looking for.

I originally tried Solve[ff-.99999999999999==0,b] and the solution was undefined by Mathematica. My solution should be at about 25 nines, but that is a guess at this point, since I cannot find an exact solution. If there is a non-Mathematica process that could be used, I am not familiar with it.

Luther

Have you tried taking a limit as b approaches 1 from the left side?

Ok, I might not understand the point of this discussion, but you appear not to be interested in

Limit[(1 + b)^2/Sqrt[1 - b^2], b -> 1, Direction -> "FromBelow"]

or

Limit[(1 + b)^2/Sqrt[1 - b^2], b -> 1, Direction -> "FromAbove"]

which is what Daniel suggests early on. Here is a different way of writing what Daniel has done in a much shorter way. First, I try to figure out for which "b" I get a certain value:

$MaxPrecision = 550    
sols = Solve[ff == c, b]

The first solution is the real valued solution. Now you can use:

sols[[1]] /. SetPrecision[c -> 9999999999999999999999999, 450]

and obtain something identical (plus/minus a couple of digits) as the simpler code before:

NSolve[ff - 9999999999999999999999999.`450 == 0, b]

You can get really close to the pole:

$MaxPrecision = 950
sols[[1]] /. SetPrecision[c -> 9999999999999999999999999999999999999999999999999999, 850]

I do not really have an easy means of verifying these numbers though. How close to the pole do you want to go?

There are not many applications, I am aware of, that need to such a high precision though, so I would like to hear of yours. Always nice to be able to tell students why one would need these high precisions "in real life".

Best wishes,

Marco

Oops, sorry, had not seen that.

BW,

Marco

PS: I'd still like to see the "application of this".

Thank you. This is another approach I had not considered. The issue was about forcing the precision to be high enough that Mathematica actually gave me an answer. My post above indicates I was looking for the radiation pressure on a large object moving at light speed. The issue was, when does the up-shifted radiation cause sufficient radiation pressure to force the object to come to a terminal speed...assuming some constant thrust or driving force. The answer is that for a massive object, depending on the driving force, the object must reach ~25 nines the speed of light to reach a terminal velocity...but in the spirit of full disclosure, the object would be fried to a crisp at much lower speeds, perhaps below 0.9c.

Regards,

Luther