MATH137 Study Guide - Asymptote, Quadratic Formula
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MATH137 Full Course Notes
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Example: sketch y = x3 x2 + 1: d : {x r, let y = 0 = x3 x2 + 1. Let x = 0 = y = 0. Thus we pass through the origin: let f ( x) , no horizontal asymptotes since the deg num > deg denom. Thus we have odd symmetry. x2 + 1 ( x)3 ( x)2 + 1 x3. We use long division: x2 + 1(cid:1) x x3. F (x) = x3 x2 + 1. Thus we have an oblique asymptote at y = x. We formally check end behaviour. x (cid:18) x3 x2 + 1(cid:19) x lim x f (x) l = lim. X x2 + 1 f (x) < x x (cid:18) x3 x2 + 1(cid:19) x lim x f (x) l = lim. X x2 + 1 f (x) > x lim x : f (x) = Thus we have a critical value at x = 0. f (x) f (x)
