There is no "limp gear." Someone just made that up.
The 5R110 is really two transmissions in one case. There is a two speed in front, with a 1:1 ratio and an overdrive 0.71:1 ratio. Behind that is a three speed with 3.1:1, 1.54:1, and 1:1 ratios.
Here's how it shifts:
First gear is both units in low, 1:1 times 3.1:1 = 3.1:1 overall ratio.
Second is the front in overdrive, the rear didn't change, so 0.71:1 times 3.1:1 = 2.2:1 second gear ratio.
Third gear is the front back in 1:1 and the rear shifts to 1.54:1, so 1:1 times 1.54:1 = 1.54:1 third gear.
I'm going to skip four for a moment.
Fifth gear is normally the next gear in the pattern. This has both units at 1:1, so that's fifth gear ratio.
Sixth gear is the front unit back in overdrive, so 0.71:1 times 1:1 = the overdrive ratio of 0.71:1.
Fourth gear was never intended to be used. It is the front in overdrive and the rear in 1.54:1 ratio, so 0.71:1 times 1.54:1 = 1.09:1 ratio. This is so close to 1:1 that it doesn't make sense to use it in the normal pattern.
When the trans is warmer than 0F it shifts 1-2-3-4-6. Some aftermarket programmers chose to make their units read 5 when they know very well the trans software calls it 6. Maybe they thought their customers were not smart enough to realize this? I don't know.
During the development of the 5R110W we found that the 3-5 shift was nearly impossible to complete when the trans was colder than 0F. We could to the 3-4, and then the 4-6 with no problems, but for reasons I won't go into here the 3-5 was nearly impossible. So we changed the shift pattern when colder than 0F to shift 1-2-3-4-6.
Later on we realized the there is a small window of speed where if you are in sixth gear and floor the go pedal you're just a bit too fast to drop back to third gear, but fifth gear doesn't quite get the acceleration that you want. In this case it actually shifts to fourth gear instead of fifth gear. This gives 9% more torque to the wheels. From fourth gear it can only shift to sixth gear or to third gear. It will NEVER shift 4-5 or 5-4.
And that's my perspective from being a calibration engineer that actually developed the Ford program that runs this transmission.