Error computing limits as x -> inf

[dbbotkin]

The problem isn't just the limit; the magnitude of the exponents blows past the calculation ability of 64bit registers. I tried scaling back the exponents to 19, 7 and 19, 6, and the CAS gives the expected 9/7 answer for Lim ∞.

"Infinity is a very long time, particularly near the end" (Woody Allen)

Other than to make a point, what is the rationale for the magnitude of the exponents? I mean, if it converges to 9/7 at E26, why expect things to change much "near the end"?