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Find the constants a and b such that lim_(x→∞) ((x²+1)/(x+1)-ax-b)=0. Which pair of values satisfies this condition?

1. a = 0, b = 0
2. a = 1, b = -1
3. a = -1, b = 1
4. a = 2, b = -1

Correct answer: a = 1, b = -1

Solution

(x^2+1)/(x+1) = x - 1 + 2/(x+1). For (expression - ax - b) -> 0 as x -> infinity, the leading and constant terms must cancel: a = 1, b = -1 (the remainder 2/(x+1) -> 0).

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