Problem

For \(f(x)~=~(x~-~1)/(x^2~-~1)\), factor and simplify while retaining every original exclusion. Classify all holes and vertical asymptotes, and state the end-behavior asymptote.

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Algebra: (x−1)/[(x−1)(x+1)]. Original exclusions: x≠−1,1. Cancellation/simplification: retain both denominator factors. Discontinuities: vertical asymptotes x=−1 and x=1; no hole. End behavior: horizontal asymptote y=0.

Algebra: (x−1)/[(x−1)(x+1)]. Original exclusions: x≠−1,1. Cancellation/simplification: cancel both denominator factors to get 1. Discontinuities: holes at x=−1 and x=1; no vertical asymptote. End behavior: horizontal asymptote y=1.

Algebra: (x−1)/[(x−1)(x+1)]. Original exclusions: x≠−1,1. Cancellation/simplification: 1/(x+1), restoring x=1. Discontinuities: no hole; vertical asymptote x=−1. End behavior: horizontal asymptote y=0.

Algebra: (x−1)/[(x−1)(x+1)]. Original exclusions: x≠−1,1. Cancellation/simplification: 1/(x+1), retaining both exclusions. Discontinuities: hole (1,1/2); vertical asymptote x=−1. End behavior: horizontal asymptote y=0.

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For \(f(x)~=~(x~-~1)/(x^2~-~1)\), factor and simplify while retaining every original exclusion. Classify all holes and vertical asymptotes, and state the end-behavior asymptote.

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