Q: What is the maximum distance the signal of the router travel to a computer ?

10-100 ft

Over ethernet it is 100M

If you are talking about wireless it is dependent on too many factors to give an answer.. in open air you can do >100M no problems. In a house, single storey, with light timber construction and no furniture it should go right through the house. Double storey or add basement or attic much harder. The airports are designed for max signal on the same plane as the airport not so much above and almost nil below. Placement becomes crucial.

Heavy walls and metal in the construction, even just alfoil insulation will cause massive wireless problems. Especially 5ghz has poor range through solids.

Then in apartment type situation you can add interference from neighbours that can be severe.

In the end trial and error is the only way.. if the range is inadequate run ethernet and further AP.. or powerline adapters or the worst possible solution.. wireless repeaters.

So do you think that I need a wireless extender?

Have you tested by wireless in the same room as the Airport.. and got your full 100Mbps download speed?? That is the first test.

But when I try to connect internet in the room of my house  the speed is Download : 2.38 Mbps.

What is the link speed of the wireless as shown by computer.

Just option click the wireless fan and check.

Speed depends on distance and number of walls etc in the path.

Here is the wireless to router 2 feet away from the computer.

Here is wireless at 10 feet through one hollow wall.

Here is the wireless to a router on the other side of the house 60 feet two brick walls that is connected by ethernet to main router.

So it goes from 1300, to 450 to 13Mbps.

These are only link speeds.. actual transfer speed is max half that. and at worst as in the case of the unit on the other side of the house.. (signal is -82 and noise is -84 that is guaranteed trouble) .. it will continually drop in and out. And could drop to speeds a tenth of link.

Using wireless extenders is the worst possible method to extend wireless.. (repeater is normal terminology for extender which is apple speak).

Use ethernet or powerline adapters (always unknown results until tested).. if you want speeds you can rely on.