Rewrite rational expressions using inspection, polynomial division, or technology.
Identify the horizontal or slant asymptote from a division result
Problem
Determine the asymptote indicated by \((x^2~+~1)/(x~-~1)~=~x~+~1~+~2/(x~-~1)\).
Big Picture
What this problem is really about
In quotient-plus-remainder form, the polynomial quotient describes the large-scale trend and the proper remainder fraction measures the remaining gap. As the input’s magnitude grows, that fraction approaches zero. Therefore the graph approaches the quotient itself; its degree tells whether the resulting asymptote is horizontal or slanted.
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