Goodman blower fan turns on with cooling for 5 minutes and turns off I recently change out the motor, the circuit board, the capacitor, and a limit switch

SOURCE: Coleman furnace blower runs continuously

You should be seeing 120 volts to the igniter. Anything less requires either a replacement of the transformer, start relay or control module.

SOURCE: I have a Goodman furnace

limit switch is on the heat exchanger black plastic about 1and1/2 in. by 2 in. with 2 wires going to it , it has 2 screws holding it in take it out and check it with an ohm meter and you should not have a reading if it is good

SOURCE: Goodman furnace GMPN080-4 Getting 4 RED blinks/Open Limit Swtch

You have a limit switch located in the furnace (a little bigger than a dime and located near the center of the front panel, sometimes the switch is mounted to a small fiberglass board) . There will be two wires going to it. They sometimes trip so often that they fail to automatically reset(due to dirty filters or undersized ductwork). You can SHUT OFF POWER TO THE FURNACE, then remove the two screws holding the switch in the furnace, pull the switch out of the furnace and tap on it and wipe it off.
The switch may work a few more times but should be replaced with the same size and type. You will see some rating printed on the switch somewhere. Write down all the numbers and get a replacemet from a furnace/AC supply house. If you have a meter and know how to use it you could check the switch for continuity.

Sometimes just a tap on the switch will get it going. There will also be a couple of rollout switches that could be tripped. They have little buttons on them to reset them. If they trip it may have other issues to be solved.
Hope this helps!
Matt

SOURCE: I have a goodman GMP 125-5 furnace. I have

sounds like the primary limit typically behind the gas valve its larger then the others, rectangle in shape with 2 wires going to it. be careful if your bypassing limit this can create a very hazardous condition

SOURCE: Four flashes on Goodman GDS80904BX circuit board

I have this clunker-4 flashes is an open high limit code according to the wiring diagram and fault code sheet. the common denomonator for this furnaces problems is the circuit board. Always flashing 3, 4, 7 times most of the time having nothing to do with the problem. The actual problem is the board it's self. take it off the mounts (POWER OFF FIRST AND SWITCH OFF>)
remove from the 4 mounts one at a time then check the back of the board. you will probably notice brown spots on the green. this is where the board has burned up. replace the board and you will be fine, you will also be the 100th person I know to have this problem. Make sure you take a picture of the board before you take all the wires off so you can put them back properly. I personally have replaced the board twice the first replacement I purchased (warranty) was made by the same company as the original, the second board I replaced 1 year later with a new board made by a new manufacturer. they obviously knew there was a problem.

http://hphotos-snc3.fbcdn.net/hs066.snc3/13331_1198507328519_1403584043_30566732_3667503_n.jpg

this is a picture of a board another fixya user sent me today after he had the same problem as you and I told him what to look for. look familliar?