Is it worth it for an eighty-seven year old alzheimer's patient to have surgery to clear a blocked carotid artery?

No 1 answer fits all: Every surgery has risks and benefits. Each specific person and their particular health history also bring benefits and risks to the discussion. Answering this question for any particular person is best done with honest open communication discussing all of the risks and benefits between the healthcare team and the person most effected.

Age & Surgery: Every operation is approached with consideration of the risks vs. Benefits. We are also taught to consider one's physiologic age rather than any set number. As a surgeon, i would have a hard time offering elective surgery to someone with (advanced) alzheimer's disease unless i knew it would improve their quality of life.

Ethical issues: From one vantage point, withholding a common procedure because of a disability is bad. However, any intervention with a level of risk must be worth while to the quality of life of the patient. If it just prolongs the process of dying, without an improvement of life, it is unfair to all involved.