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Find the constants a and b such that \[\lim_{x\to\infty}\left(\frac{x^2+1}{x+1}-ax-b\right)=0.\] 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

To satisfy the limit condition, we need to ensure that the expression approaches zero as x approaches infinity. By substituting a = -1 and b = 1, we simplify the expression to a form that cancels out the leading terms, resulting in a limit of zero.

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