MATH 12002 - CALCULUS I 1.6: Vertical & Horizontal Asymptote - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I § 1.6: Vertical & Horizontal Asymptote Examples Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 6

Examples Example Find all vertical and horizontal asymptotes for the function f ( x ) = 3 x 2 + 2 x + 7 2 x 2 − 8 x − 10 . D.L. White (Kent State University) 2 / 6

Examples Solution Horizontal Asymptotes: Since f ( x ) is a rational function with numerator and denominator of the same degree, the horizontal asymptote is the quotient of the leading coefficients; that is, y = 3 / 2 . Vertical Asymptotes: The denominator of f ( x ) is 2 x 2 − 8 x − 10 = 2( x 2 − 4 x − 5) = 2( x + 1)( x − 5) , which is 0 when x = − 1 or x = 5 . When x = − 1 , the numerator is 3( − 1) 2 + 2( − 1) + 7 = 3 − 2 + 7 = 8 � = 0 and when x = 5 , the numerator is 3(5) 2 + 2(5) + 7 = 75 + 10 + 7 = 92 � = 0 . Hence the vertical asymptotes are x = − 1 and x = 5 . D.L. White (Kent State University) 3 / 6

Examples Example Find all vertical and horizontal asymptotes for the function 9 − x 2 f ( x ) = 3 x 2 − 3 x − 18 . D.L. White (Kent State University) 4 / 6

Examples Solution Horizontal Asymptotes: Again f ( x ) is a rational function with numerator and denominator of the same degree, and so the horizontal asymptote is the quotient of the leading coefficients; that is, y = − 1 / 3 . Vertical Asymptotes: Observe that 9 − x 2 3 x 2 − 3 x − 18 = (3 + x )(3 − x ) f ( x ) = 3( x + 2)( x − 3) , and so the denominator is 0 when x = − 2 or x = 3 . When x = − 2 , the numerator is not 0 , hence x = − 2 is a vertical asymptote. [Continued → ] D.L. White (Kent State University) 5 / 6

Examples Solution [continued] However, when x = 3 , the numerator is also 0 , and in fact (3 + x )(3 − x ) x → 3 f ( x ) lim = lim 3( x + 2)( x − 3) x → 3 − (3 + x )( x − 3) = lim 3( x + 2)( x − 3) x → 3 − (3 + x ) = lim 3( x + 2) , since x � = 3 , x → 3 − (3 + 3) = lim 3(3 + 2) x → 3 − 6 15 = − 2 = 5 � = ±∞ . Therefore, x = 3 is NOT a vertical asymptote. D.L. White (Kent State University) 6 / 6