Problem

For \(x~>~0\), interpret the quotient trend and correction term in \(C(x)~=~(x^2~+~5x~+~6)/x~=~x~+~5~+~6/x\). State the asymptote and the direction of \(C(x)\)'s approach.

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What this problem is really about

Separate the model into a quotient trend and a remainder correction. The correction’s magnitude tells how far the function is from the trend, its sign tells which side of the trend the graph occupies, and its limit tells whether that gap closes. Use the stated domain when determining the sign, since approach direction can depend on it.

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The quotient trend is y=x+5. The correction 6/x is treated as zero for every x>0, so C(x) equals the trend exactly rather than approaching it.

The quotient trend is y=x+11. The correction is 6/x for x>0, so C(x) lies above that trend and the difference approaches 0 as x increases.

The quotient trend is y=x+5. The correction is -6/x for x>0, so C(x) lies below the trend and the difference approaches 0 as x increases.

The quotient trend is y=x+5. The correction 6/x is positive for x>0, so C(x) lies above the trend and the difference approaches 0 as x increases.

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For \(x~>~0\), interpret the quotient trend and correction term in \(C(x)~=~(x^2~+~5x~+~6)/x~=~x~+~5~+~6/x\). State the asymptote and the direction of \(C(x)\)'s approach.

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