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

10.3 Limits and continuity 361
Thelimitofafunction, atapointx =a, existsonly iftheleft-handand right-hand
limits areequal there.
10.3.1 Continuousanddiscontinuousfunctions
Afunction f iscontinuousatthe point wherex =a, if
lim f =f(a)
x→a
that is, the limit value matches the function value at a point of continuity. A function
which is not continuous is discontinuous. In Example 10.3, the function is continuous
atx = 0 because
lim y=0=f(0)
x→0
butdiscontinuousatx = 2becauselim x→2
ydoesnotexist.InExample10.2,thefunction
isdiscontinuousatx = 0becauselim x→0
y = 1buty(0) = 3.Theconceptofcontinuity
corresponds to our natural understanding of a break in the graph of the function, as
discussed inChapter 2.
Afunction f iscontinuous atapointx =aifand only if
lim x→a
f = f(a)
thatis, the limitof f exists atx =aand isequal to f (a).
EXERCISES10.3
1 Thefunction, f (t),is defined by
⎧
⎨1 0t2
f(t)= 2 2<t3
⎩
3 t>3
Sketchagraphof f (t)andstatethefollowinglimitsif
they exist:
(a) lim t→1.5 f
(b) lim t→2 + f
(c) lim t→3 f
(d) lim t→0 + f
(e) lim t→3 − f
2 The functiong(t)is defined by
⎧
0 t<0
⎪⎨
t
g(t) =
2 0t3
2t+3 3<t 4
⎪⎩
12 t>4
(a) Sketchg.
(b) State any pointsofdiscontinuity.
(c) Find, ifthey exist,
(i) lim t→3 g
(ii) lim t→4 g
(iii) lim t→4 −g
Solutions
1 (a) 1 (b) 2 (c) notdefined
(d) 1 (e) 2
2 (b)t=4
(c) (i) 9 (ii) not defined (iii) 11