When tranny try to shift into 3th gear(no pull power),it slips 1,2 and reverse works OK.

It may just need servicing/adjusting.

Your reverse gear is actually first gear ... when you select reverse another 'cog' fits into the toothed wheels to change the direction of first gear ... to give you reverse...

It may be a sticky valve or just a bit of adjustment (workshop job..) that is required.

What you can try ..

Start the car ... move the lever through all the positions several times (foot on brake..). Keep your foot on the brake and then put it in reverse and rev the car. Hard!

Try doing it several times.

An auto transmission works on valves opening and closing. Reverse gear - in comparison with the forward gears - is seldom used. The reverse valve can 'stick'. Try forcing it to open ... frighten it ... but keep your foot on the brake...

Look at the shift solenoid or oil screen/filter. Oil bad need flush and new oil. Over heater clogged oil cooler. Burned tranny clutches. Start on the oil

Trans internal failure.

that is a hum dinger, largest ever posted, in one shot.
very good.!
one word , the transmission slips. (never drive a slipping tranny)
overheat indicator.! that is key.
many newer cars will change the transmission shifting any time
the engine over heats.
and overheat wins over transmission , every time.
if the engine burns up, what good is any transmission.? nada.

solve overheat first.

slip or flare.
is when the engine RPM rises fast and the car dont go faster.
SLIPS.
and slipping burns up trannyclutchs and increases the damage. inside the transmission, dont do that. dont allow it,
park it and fix it.

on some tranny there is a temperature sensor that the TCM
watches and changes shift patterns based on AFT fluid temps.
and if the ATF fluid gets too hot , it takes actions. to keep the tranny from harm.

why not get a scan too and see what DTC errors there are..
DTC, engine overheate.
DTC< tranny overheat (caused by above)

can i assume you pulled the side plug on tranny and the level
is right there? or gone way down, ever do a 50k service? to it?

M/T tranny (not auto) what engine? the m./t varies by HP of ENGINE.
If its really in 5th ( i bet not)
if the RPM rises fast that is called clutch slip
yes, clutch bad.
if not in 5th actual rpm will rise fast too. slip,
if the other gears wroke like 4th at wide open throttle up hill
gee the chutch is good.
so its not going in to 5th (for lots of reasons)
but 1 reason , is easy.

-AX 4/5/15 transmission
this tranny has a shift tower. (that thing on top that the shifter lever
fits in to)
most tranny that fail to actual shift into gear
is caused by a $1 plastic part failure
so pull the shifter off top of the shift tower.
and look, is the plastic buffer parts all bad, missing, cracked or
destoryed, sure they all end there... all.
i can also remove the shifter, lever.
and shift to 5ht by hand, and see that it DOES.
it either does, or does not.
if it does, the lever parts are bad.
that simple.
see part 61 (im guessing which of 5 tranny it is , )
http://cdn.liveoffroad.com/images/Aisin_AX5_Transmission.jpg

Check your input or output sensors on top of tranny. Check engine light?

don't want to shift? fills a huge basket ,
1: grinds?
2: can't get in to some gears.? or all gears.? (engine running and not)?
3: clutch drags, clutch slips kills all gears, or just 5th.
drag kills them all and reverse 1st , its the canary gear. !! non synchro
4: noise.? or chatter, ****,larchig, or throwout hash ,?

here is the clutch page, link below.

the #1 failure for a gear not Selecting is a $5 fix.
the tip of the shifter shaft, plastic POS is broken, bar none this failure,#1 and sold only at SUZ, get em' while you can they are chapter 11 ANDGM/GEO is brain dead , ask them , and you get blank stares.

this can be proven, by selecting the gears by hand. running.
if failure, cant move those fork rods?
running? only
if you cant, the clutch is dragging. see why?
works stalled means the clutch drags.
can you shift to 1st stalled, start engine drive off can you get any gear? yes,if yes , constantly? with no grinding, the clutch is ok.

to do that manual shifter off test test the shifter must be removed.



the we get to the cable.
see the 1inch rule here, no guessing.
play set to spec, and you floor the pedal , person 2 sees 1 inch travel, (END new cable worry)
if it moves 1inch and the tranny can not be SELECTED.
the clutch is bad, if you can select only stalled motor.
if selection fails all time, stalled and running, the tranny is siezed.
covers every fail logged by 1000s of users

http://www.fixkick.com/tranny/clutch/clutch-on-kick.html

Could be a long list of things.
Its too late for a filter change. Doing that now would probably make matters worse.
The trans uses electric solenoids and a pump to apply pressure to the clutches. So either could be faulty given the age and mileage.
I'm not sure if the shifting problem is connected to the shift indicator lights problem.
You may need to take it to a trans shop to find out for sure. And they will not want to touch it with that many miles on it.
A regular mechanic might be able to do a pressure test on it to at least find out if it is the pump or not.

Worn clutch plates. Reverse & first gear uses the same clutches so how is 1st gear running out as you drive off? Slips a little?

Sir I would first change my filter. When you do you should be able to acertain if anything is wrong by what yuo find in your pan. I think your clutchs are probably bad, but with out being able to scan it I don't have any other ideas. I have been lucky a couple of time before and changing the filter solved my problem. Your car is in limp mode generally you have reverse and second gear. 1st gear and reverse use the same clutches, so I am wondering if maybe you have second gear only!