atanh

I was a little startled to find today that atanh's finite values seem to be bounded roughly between -18.71497 and 18.71497. These correspond to inputs of -0.999999999999999944444... and 0.99999999999999994444444... Stepping just a bit beyond these (changing any of those 4s to a 5) we get back an infinite value. I suppose I should work out analytically what's going on. Or just look at this:

> q <- function(x) (1+x) / (1 -x)
> q(0.99999999999999995)
[1] Inf
> q(0.99999999999999994)
[1] 1.80144e+16

The 9899 standard just says, "The atanh functions compute the arc hyperbolic tangent of x. A domain error occurs
for arguments not in the interval [-1, +1]. A range error may occur if the argument equals −1 or +1." Where should I look for more details?

Using R's pnorm I see there's roughly probability 1.869160e-78 beyond these points in the tails of a standard Gaussian distribution (and so the probability of seeing the points beyond them is indistinguishable from zero for any variances much less than one) so that this is of little importance for Fisher's z transform 0.5 atanh(r). Still it's a surprise to me not to see a smooth transition.

> atanh(tanh(18.71497))
[1] 18.71497
> atanh(tanh(18.71498))
[1] Inf