082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

76 Chapter 2 Engineering functions
y
y
y
3
1
x
–2
x
–1
x
(a)
(b)
Figure2.22
Some examples offunctionswith their asymptotes:
(a)y= 3x+1
x
= 3 + 1 −1 +4x+2
;(b)y=x
x x +2 ;(c)y=x2 =x+3− 1
x +1 x +1 .
(c)
and the y axis, that is x = 0. In Figure 2.22(b) the asymptotes are the horizontal line
y = 1andtheverticallinex = −2;inFigure2.22(c)theyarey =x +3andthevertical
linex = −1.Theasymptotey =x+3,beingneitherhorizontalnorvertical,iscalledan
oblique asymptote. Oblique asymptotes occur only when the degree of the numerator
exceeds the degree of the denominator by one.
Weseethattheverticalasymptotesoccuratvaluesofxwhichmakethedenominator
zero. These values are particularly important to engineers and are known as the poles
of the function. The function shown in Figure 2.22(a) has a pole atx = 0; the function
showninFigure2.22(b)hasapoleatx = −2;andthefunctionshowninFigure2.22(c)
has a pole atx = −1.
Ifthegraphofafunctionapproachesastraightline,thelineisknownasanasymptote.
Asymptotesmaybehorizontal, vertical oroblique.
Values of the independent variable where the denominator is zero are called poles
ofthe function.
Example2.10 Sketch the rationalfunctiony =
x
x 2 +x−2 .
Solution For large values ofx, thex 2 term in the denominator has a much greater value than the
x inthe numerator. Hence,
y→0 as x→∞
y→0 as x→−∞
Therefore thexaxis, thatisy = 0,isanasymptote. Writingyas
x
y =
(x −1)(x +2)