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If lim(x -> infinity) [ (x³ + 1)/(x² + 1) - (a*x + b) ] = 2, then the values of a and b are

1. a = 1, b = 1
2. a = 1, b = 2
3. a = 1, b = -2
4. none of these

Correct answer: a = 1, b = -2

Solution

Divide x³+1 by x²+1: x³+1 = x*(x²+1) - x + 1. So (x³+1)/(x²+1) = x + (-x+1)/(x²+1). Therefore the expression = x + (-x+1)/(x²+1) - ax - b = (1-a)x - b + (-x+1)/(x²+1). For the limit as x->inf to equal 2 (finite), the coefficient of x must be zero: 1-a=0 => a=1. Then the limit = -b + lim[(-x+1)/(x²+1)] = -b + 0 = -b = 2 => b = -2.

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