The only sure way to determine the fiber content of a fabric outside a lab is with a burn test. (a burn test is also used in the lab and with experience can be pretty accurate.)

Take a small yarn from the fringe or some frayed area.
Hold the fiber in tweezers.
Hold a match to it and watch how it reacts to the flame.
Then hold it to your nose and smell.

Compare what you see and smell to this chart.



Basically, if the yarn burns and does not self extinguish leaving a small, hard, ball of melted plastic and smells like a barbeque you have Acrylic.

If the yarn does not stay burning and leaves a crumbly, gray ash, and smells sharp like paper or chemicals you have Polyester/Rayon or P/V.

Wool will not stay lit at all, will leave a dry, light gray ash, and smells like what it is, burning hair.