A photo would help.
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Photos would help greatly.

 

Sorry, I thought I attached them.

 

A couple of layers of caulk. Too big a gap to fill in one shot
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What are your thoughts on cutting 1/2 inch slivers of tile. The tile was installed with 1/8" grout lines. This would leave a 1/8" gap at the threshold. Do you think that would look better or worse?

 

Depending on the specific tile that you have, it may be impossible to cleanly cut off a narrow 1/2" slice. at least in one piece. If you have extra tile, sure, give it a try,. But I think caulk would look less noticeable.
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To fill the gap I would try to add an aluminum (1/2" or 5/8" wide) U-channel next to the door threshold.
The legs of a channel can be trimmed on the table saw to align with the door threshold height

 

I would probably use something like this. It would actually end up looking pretty good I think. If you tried to fill in tile there it would look like amateur hour. Don't overapply or you'll have a mess.


https://retail.usa.sika.com/en/products/sealants/self-leveling/sikaflexr-self-leveling-sealant-0

 

I initially saw just the darker gap and thought...gosh that's not much wider than a grout line. I think the gap is the white and darker area? If so I would definitely first try cutting some thin pieces of tile. I did it as I had an area that wasn't parallel to the wall and so I needed just about 1/2" tapered slices. I had to cut carefully but I ended up with some good pieces and it looks great. If that last tile is already a cut piece it will look a bit less good but I disagree with anyone here that says putting in a small tile is a more janky solution than filling it with 3/4" of caulk. I don't see how that will possibly look better than the tile option.

 

If that last tile is already a cut piece it will look a bit less good but I disagree with anyone here that says putting in a small tile is a more janky solution than filling it with 3/4" of caulk. I don't see how that will possibly look better than the tile option.

The stuff I linked doesn't look like standard caulk exactly, but the problem with the tile slice is that it looks like you installed the tile all wrong - from back to front. In other words, like you didn't know what you were doing. The self leveling compound actually has a reason for being there, as that is in a sense an expansion joint of sorts.

 

I think the tile was the right choice. It's barely even noticeable.