Maybe there is a life limit to it, just like anything els.
But I would take it out of the pot, examin the roots. If there are dead or decaying root, prune them with sharp knife. Also , possibly divide and repot it. If you have not done this before, probably the pot is jammed with lots of roots.

11 years does seem quite a long time to have kept the same plant in a pot - have you repotted it at all? If not, then it could be that it's so full of roots there's no space for it to produce new ones.

You could tip it out the pot as cyrus gardener suggests and if the roots seem healthy, tease out and replace as much soil as you can. I'd cut out all the dead bits and also try taking a few cuttings in case you lose it.

Unless you're emotionally attached to it, perhaps it would be best to replace it with a new one?

Thanks for the followups.... I have been taking it out of the pot and cutting back the roots every other year. I am feeling around the dirt in the container, and the roots definitely are not rootbound. In fact, there doesn't seem to be much resistance at all, which makes me think the roots are dead...I'll take it out of pot and examine. Yes, after 11 years, I am attached to plant, I'll do what I can to save it if possible, then it's time to buy a new one.

You didn't answer patsy's question about repotting. In those 11 years, have you changed the potting soil frequently? Root pruning, alone, wouldn't be be sufficient. Potting mediums have a finite life span, too....just like everything else.

I don't know the life expectancy of rosemary, but there are many planted on a university campus in my city that are alot older than 11 years. But they are in the ground.

The webpage below says rosemary can live for 30 years. Seems reasonable for a woody shrub. But I would suspect that a container plant would have a shorter life span than one in the ground outdoors in the warm embrace of Mother Nature - but that's my opinion.

And yes, you do get attached to those plants you've had for a while. I've got a norfolk pine I've had for 35 years and 2 plants that were my grandmother's and are about as old. I would feel bad to loose "Peter Pine" and the others. I hope your rosemary survives for you!

FataMorgana

Here is a link that might be useful: Rosemary

Yes..I added new soil to the container 2 years ago (about 1/2 the container). What do you think about the drastic pruning?

Before you try to save it, you need to figure out why it's not doing well. Is it possible some liquid that wasn't water accidentally got dumped into the pot (party guests can sometimes empty drink glasses inappropriately!)? Was there poor drainage from overwatering, or a buildup of fertilizer salts in the pot that need flushing out? Was it located where there's far too little sunlight or too near a heating source? (just stabs in the dark here; I live in a Mediterranean climate and only grow it outdoors--but one, in a strawberry pot, periodically stresses out because I don't water it enough. It doesn't die, though, has never been repotted because there's no way to get it out of the container without breaking the pot, and it's at least as old as yours.)

I'd prune off the part that's definitely dead, but wouldn't prune the rest drastically if the plant is already stressed, especially if any of the above are possible reasons it's not doing well. First, you'd want to correct those, and see how it responds.

In the meantime, though, rather than wait for it to do even more poorly, take some cuttings and root them (rosemary is very easy to propagate this way). That way, at least you'll be extending the life of the original plant.