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

▶

$math 12002 calculus i 1 6 vertical horizontal asymptote$

Mar 21, 2024 13 likes •76 views

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

SLIDE 1

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

SLIDE 2

Examples

Example

Find all vertical and horizontal asymptotes for the function f (x) = 3x2 + 2x + 7 2x2 − 8x − 10.

D.L. White (Kent State University) 2 / 6

SLIDE 3

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 2x2 − 8x − 10 = 2(x2 − 4x − 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

SLIDE 4

Examples

Example

Find all vertical and horizontal asymptotes for the function f (x) = 9 − x2 3x2 − 3x − 18.

D.L. White (Kent State University) 4 / 6

SLIDE 5

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 f (x) = 9 − x2 3x2 − 3x − 18 = (3 + x)(3 − 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

SLIDE 6

Examples

Solution [continued]

However, when x = 3, the numerator is also 0, and in fact lim

x→3 f (x)

= lim

x→3

(3 + x)(3 − x) 3(x + 2)(x − 3) = lim

x→3

−(3 + x)(x − 3) 3(x + 2)(x − 3) = lim

x→3

−(3 + x) 3(x + 2) , since x = 3, = lim

x→3

−(3 + 3) 3(3 + 2) = −6 15 = −2 5 = ±∞. Therefore, x = 3 is NOT a vertical asymptote.

D.L. White (Kent State University) 6 / 6