Consider F(x)=[x^4 - x^2 + 1]/[x^4 + 2x^3 + x] for x not = 0
F(x) tends to 1 as x tends to infinity. When x is large, f(x) is approx but not exactly 1
Use change of variable y=1/x then do a maclaurin expansion in y to show that f(x) is approx 1-2x for large x and find the third term
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