Problem

Compare the vertical and horizontal asymptotes and end behavior of \(A(x)~=~8~+~500/x\) and \(B(x)~=~10~+~200/x\). State the limiting value of \(B(x)~-~A(x)\) as \(|x|\) grows.

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What this problem is really about

Analyze the two rational models in layers. The reciprocal term controls the shared vertical asymptote and the one-sided behavior near it, while the constant term gives each horizontal asymptote as the input grows in magnitude. Subtracting the functions is the cleanest way to see what their vertical gap approaches far from the asymptote.

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Both have vertical asymptote x = 0. A approaches y = 8 at both ends, while B approaches y = 10; their long-run difference B − A approaches 2.

Both have vertical asymptote x = 0. A approaches y = 10 at both ends, while B approaches y = 8; their long-run difference B − A approaches 2.

Both have vertical asymptote x = 8. A approaches y = 8 at both ends, while B approaches y = 10; their long-run difference B − A approaches 2.

Both have vertical asymptote x = 0. A approaches y = 8 at both ends, while B approaches y = 10; their long-run difference B − A approaches −2.

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Compare the vertical and horizontal asymptotes and end behavior of \(A(x)~=~8~+~500/x\) and \(B(x)~=~10~+~200/x\). State the limiting value of \(B(x)~-~A(x)\) as \(|x|\) grows.

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