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How to find the horizontal asymptotes of a rational function by evaluating the limit of the function as x approaches infinity. It includes examples and general rules for polynomial functions.
Typology: Exercises
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In order to determine the horizontal asymptotes of a rational functionthe following limit: lim R(x) = f g^ ((xx)) we must evaluate xโโ R(x). Example: Evaluate lim xโโ^2 x^2 x^ + (^2) + 1^ x^ โ^1 Solution: Notice that 2the numerator and denominator byx^2 + x โ 1 โ โ 1 and x^2 + 1 โ โ as x โ โ. For this function we can multiply x^2 to determine the limit. xlimโโ^2 x^2 x^ +^2 + 1^ x^ โ 1 = lim xโโ^ (2x^2 +^ x^ โ^ 1)
( (^) x 12 ) (x^2 + 1)^ (^ x^12 )^ = lim^ xโโ
2 +^1 x โ (^) x^12 1 + x^12 =^ xlimโโ^ 2 + lim x limโโ^ x 1 + limโโ x^1 xโโ^ โ^ x xlim 1 โโ 2 x^12 = 2 + 01 + 0^ โ 0 = 2 Here we are exploiting the fact that (^) xlimโโ x^1 n = 0, where n > 0 is a rational number. Therefore f (x) =^2 x^2 x^ + (^2) + 1^ x^ โ 1 has a horizontal asymptote along the line y = 2 as x approaches โ. Example: Find the horizontal asymptotes of g(x) = (^) x (^2) + 3x^ + 1x โ 4
xlimโโ x^2 + 3^ x^ + 1x โ 4 = lim xโโ^ (x^ + 1)
( (^) x 12 ) (x^2 + 3x + 4)^ (^ x^12 )^ = lim^ xโโ
(^1) x + x (^12) 1 + x^32 โ (^) x^42 = x limโโ 1 + lim^ xlimโโ xโโ^1 x^ + lim x 3 x 2 โโ โ (^) x^1 xlimโโ x 42 = (^) 1 + 00 + 0 โ 0 = 0 We conclude that g(x) has a horizontal asymptote along the line y = 0. Noteby 1 : In general for polynomial functions f (x) and g(x) we multiply the numerator and denominator xm^ where^ m^ is the highest power of^ x^ that appears in the denominator.
Given the rational function R(x) = f g^ ((xx) )= (^) bamnxxmn^ ++^ abnmโโ^11 xxnmโโ^1 1 + (^) +^ ยท ยท ยท ยท ยท ยท^ + +^ a b^11 xx^ + +^ a b^00 , the horizontal asymptotes for R(x) are determined by the following cases: (i) If deg(f (x)) > deg(g(x)), R(x) has no horizontal asymptote. (ii) If deg(f (x)) = deg(g(x)), R(x) has a horizontal asymptote at y = (^) bamn. (iii) If deg(f (x)) < deg(g(x)), R(x) has a horizontal asymptote at y = 0. Note: deg(f (x)) and deg(g(x)) represent the degree of f (x) and g(x), respectively. Examples (i) R(x) = x x^32 + 1โ 1 , deg(f (x)) > deg(g(x)). R(x) has no horizontal asymptote.
(ii) y = R 4 (x) =^4 x^22 โx (^2 3) + 7x^ + 1 , deg(f (x)) = deg(g(x)). R(x) has an horizontal asymptote along the line 2 = 2. (iii) R(x) = (^) xx (^2) + 5+ 1x, deg(f (x)) < deg(g(x)). R(x) has an horizontal asymptote along the line y = 0.