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

360 Chapter 10 Differentiation
whenxisnear,butdistinctfrom,0?’FromFigure10.6weseeyisnearto1,thatis
lim y = 1
x→0
Example10.3 The functiony(x) isdefined by
⎧
⎨0 x0
y(x) = x 0<x2
⎩
x−2 x>2
(a) Sketch the function.
(b) State the limitofyasxapproaches (i)3,(ii)2,(iii)0.
Solution (a) ThefunctionisshowninFigure10.7.Notethatthefigurehasthreeparts;eachpart
corresponds toapartinthe algebraic definition.
(b) (i) Asx → 3,the relevant partof the function isy(x) =x −2.Hence
lim y = 1
x→3
(ii) Supposex < 2 and gradually increases, approaching the value 2. Then, from
the graph, we see that y approaches 2. Now, suppose x > 2 and gradually
decreases,tendingto2.Inthiscaseyapproaches0.Hence,wecannotfindthe
limitofyasxtends to2.The lim x→2
ydoes notexist.
(iii) Asxtends to 0,ytends to 0. This is true whetherxapproaches 0 from below,
thatisfrom the left,orfrom above, thatisfrom the right.So,
lim y = 0
x→0
y
y = x
y = x – 2
y = 0
2
x
Figure10.7
Thefunctionyhas different limitsasx → 2
from the left and the right.
Itisappropriateatthisstagetointroducetheconceptofleft-handandright-handlimits.
ReferringtoExample10.3,weseethatasxapproaches2fromtheleft,thatisfrombelow,
then y approaches 2. We say that the left-hand limit of y as x tends to 2 is 2. This is
written as
lim
x→2 −y = 2
Similarly, the right-hand limit ofyis obtained by lettingxtend to 2 from above. In this
case,yapproaches0.This iswritten as
lim
x→2 +y = 0
Consider a point at which the left-hand and right-hand limits are equal. At such a point
wesay ‘thelimitexists atthatpoint’.