Will using plus or premium gas in my 2001 Altima improve its performance?

My product manual for my 2001 Nissan Altima says I should use "at least" 87 for the car's fuel. Does this mean that I could use a higher octane like 89 or 93? If so, would I get better performance out of the car by using a higher fuel grade? Could it damage the car if I did this? And finally, is... show more My product manual for my 2001 Nissan Altima says I should use "at least" 87 for the car's fuel. Does this mean that I could use a higher octane like 89 or 93? If so, would I get better performance out of the car by using a higher fuel grade? Could it damage the car if I did this? And finally, is it worth the extra money to get a higher fuel grade?
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