I recently researched how much professional indemnity insurance would cost for a UK sole trader completing computing contracts overseas (specifically USA). Since the contracts fall under US law the insurance was very high and quite rubbish.
With this in mind I am considering incorporating and using the limited liability associated with an Ltd to protect my personal assets. Does Ltd limited liability apply if my business is sued under US law?
That is the intention, I believe, as long as you don't act dishonestly. However, you should bear in mind that having a Ltd company doesn't mean they can't sue you personally. Anyone can sue anyone for anything.
For the PI it is where the clients/work are based, even if the contracts are governed by English law. The USA is more litigious and it is much more expensive for a UK insurer to defend you in the US, so rates are higher.
IANAL, IMHO, YMMV, etc.
Steve Jones's answer is basically right.
The point is that if you create a Ltd company then there are two legal persons: you and the company. If "your business" is the company and it is sued, then, except where you have done various things you should not have done, all they will be able to take are the assets of the company. They aren't suing you.
In the typical case where there is a breach of contract and they sue the company, you could be reasonably confident of being safe.
But creating a limited company doesn't immunise you from being sued in the normal way. If you committed fraud, breached confidence, or defamed them (to give 3 examples) then they could sue you.
There are ways of what the courts call "piercing the corporate veil" but they will generally involve some form of personal misconduct as director or something else which goes beyond the sorts of things I'd hope you are doing in business.