LECTURE 3 OF 4 TOPIC : 8.0 LIMITS AND CONTINUITY 8.2 ASYMPTOTES
(a) Find the vertical and horizontal asymptotes OBJECTIVES At the end of this topic, students should be able to: Sketch the graph of a rational function involving the vertical and/or horizontal asymptotes. (b)
VERTICAL ASYMPTOTES ( x = a ) A line x = a, is a vertical asymptote for the graph of y = f(x) if either : x a f x i. lim ( ) ii. x a f x lim ( ) Therefore x = a is the vertical asymptote Method: i. Direct simplify ii. Factorize iii. Multiply by conjugate OR
1 f x( ) x D ( , ) ( , ) 0 0 f x 0 D R\ x 0 f OR 0 1 limx x Vertical Asymptote: (the values of x that will make the function undefined : x = 0 ) Therefore, x = 0 is the vertical asymptote 0 1 lim x x 0 x 0 f x x 0 f x vertical asymptote
HORIZONTAL ASYMPTOTES ( y = b ) A line y = b is horizontal asymptote for the graph of y = f(x) if either : x f x b i. lim ( ) ii. x f x b lim ( ) Therefore y = b is the horizontal asymptote Method: Divide with the highest power of denominator OR
1 f x( ) x D ( , ) ( , ) 0 0 f x 0 D R\ x 0 f OR Thus y = 0 is the horizontal asymptote 0 1lim 0x x 1 lim 0 x x x f x 0 f x 0 x horizontal asymptote
0 x 0 f x x 0 f x x f x 0f x 0 x x = 0 (vertical asymptote) y=0 (horizontal asymptote) 1 f x( ) x x 0 Sketch the graph of a rational function involving the vertical and/or horizontal asymptotes
EXAMPLE 1: Find the vertical and horizontal asymptotes of 2 1 f x( ) , x x 0 SOLUTION: Hence, sketch the graph of the function.
f(x) x X 0 + 2 1 ( ) x f x x x ( ,0) (0, ) Df Horizontal asymptote: GRAPH : 2 1 1 lim 0 x x The horizontal asymptote is y = 0. 2 1 1 lim 0 x x 2 0 1 1 lim x x 0 The vertical asymptote is x = 0 2 0 1 1 lim x x 0 Vertical asymptote: SOLUTION: x 0 -
EXAMPLE 2: Determine the asymptotes of SOLUTION: 2 1 ( ) 5 6 f x x x 2 1 1 ( ) 5 6 ( 6)( 1) f x x x x x x x 6& 1 ( , 6) ( 6,1) (1, ) Domain: Df Hence, sketch the graph of the function.
6 1 1 lim x ( 6)( 1) 0 x x lim 1 1 x 1 ( 6)( 1) 0 x x lim 1 1 x 1 ( 6)( 1) 0 x x Vertical asymptotes are x = - 6 and x = 1 6 1 1 lim x ( 6)( 1) 0 x x Vertical asymptotes: 2 1 lim x x x 5 6 0 1 Horizontal asymptote is y = 0 Horizontal asymptote :
y x = -6 x = 1 6 1 y = 0 x GRAPH : Vertical asymptotes are x = - 6 and x = 1 Horizontal asymptote is y = 0 ( , 6) ( 6,1) (1, ) 2 Df 1 ( ) 5 6 f x x x
EXAMPLE 3: Determine the asymptotes of SOLUTION: 2 1 ( ) 1 x f x x 2 1 1 ( ) 1 ( 1)( 1) x x f x x x x x 1 ( , 1) ( 1,1) (1, ) Df Hence, sketch the graph of the function.
x = 1 is not the vertical asymptote Vertical asymptote is x = - 1 Vertical asymptotes: Horizontal asymptote is y = 0 Horizontal asymptote : 1 1 1 lim x x 1 2 1 1 1 lim x x 1 2 1 1 1 lim x x 1 0 1 1 1 lim x x 1 0 1 lim x x 1 1 lim x 1 lim ( ) lim ( 1)( 1) x x x f x x x 0
( 1)( 1) 1 ( ) x x x f x x = -1 y = 0 -1 0 1 1 ( 1) 1 ( ) x f x y x GRAPH : Vertical asymptote : x = -1. Horizontal asymptote : y = 0 ( , 1) ( 1,1) (1, ) Df
EXAMPLE 4: Determine the asymptotes of SOLUTION: 2 2 ( ) 4 f x x 2 2 2 ( ) 4 ( 2)( 2) f x x x x Domain: ( , 2) (2, ) Df ( 2)( 2) 0 x x Hence, sketch the graph of the function.
Vertical asymptotes: Horizontal asymptote : 2 2 lim ( 2)( 2) x x x 2 2 lim ( 2)( 2) x x x Vertical asymptotes are x = -2 and x = 2 2 lim 0 ( 2)( 2) x x x 2 lim 0 ( 2)( 2) x x x Horizontal asymptote : y = 0 x=-2 x=2 -2 2 0 y=0
EXERCISE 1: Determine the asymptotes of SOLUTION: f x x ( ) ln 0 D , f R , f x y 1 0 lim ln x x Vertical Asymptote : Horizontal Asymptote : No horizontal asymptote 1 ( ) log , a X=0 is the vertical asymptote f x a x GRAPH : Hence, sketch the graph of the function.
EXAMPLE 5: SOLUTION: Vertical Asymptote : No vertical asymptote Horizontal Asymptote : 1 1 lim 0 x x e e ( ) lim x x e e e Horizontal asymptote is y=0 ( ) xf x eDetermine the asymptotes of D , f x y f (x) a ,0 a 1 x 1 GRAPH : Hence, sketch the graph. EXERCISE 2:
CONCLUSIONS ( ) lim x a ( ) f x g x ? 0 ? 0 ? 0 ? 0 Vertical asymptote : x = a (Limit is infinite) Method: i. Direct simplify ii. Factorize iii. Multiply by conjugate
CONCLUSIONS ( ) lim ( ) g x f x x f(x) and g(x) are both polynomial function, then Divide each term by x with the highest power of the denominator Method: Divide with the highest power of denominator Horizontal asymptote : y = b (Limit at infinite)