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It's most likely open diff. You can decode the door sticker to find out your gear ratio and differential type.
You can also look at the axle tag bolted to the covers (or axle beam since its TTB). If your axle ratio is 4.10 with an open diff you should see a 4 10 stamped on it. Limited slips have an "L" where the space would be.
Probably because when you're off road, limited slip is hardly an improvement over open... *almost* as crappy but has a better chance of the working if you try the parking brake trick.
I've never had a limited slip in a front axle, but I had a locker in one for a little while, and I'll never do that again in a 4wd that I plan on using on snowy highways. When only one front tire loses traction (like as you start to change lanes and the left tires hit the snow between lanes) the front tire with traction (the right in my example) not only tries to torque-steer the truck to the left like when the same thing happens with a rear locker, but also the torque steer tries to turn the steering wheel to the left as the knuckle tries to pivot on the ball joints. The result was that the truck moved 10 feet to the left before I could catch it. Fortunately the lane was empty so I didn't hit anything. That happened in the first snow storm I drove in after putting in the locker. I got off at the next exit and unlocked the hubs, then took the locker out and went back to an open front diff the first chance I got.
A limited slip would tend to do the same thing, but not quate as badly (especially if it's a really weak limited slip). But I'll never have anything other than an open diff (or a selectable locker that's open when unlocked) in the front axle of a winter street-driven truck.