5.1 math

Q1:Provethatthefunction iscontinuousat

Answer:

Therefore,fiscontinuousatx=0



Q2:Examinethecontinuityofthefunction.

Answer:

Thus,fiscontinuousatx=3

Q3:Examinethefollowingfunctionsforcontinuity.

(a) (b)

Exercise5.1:SolutionsofQuestionsonPageNumber:159

(c) (d)

Answer:

(a)Thegivenfunctionis

Itisevidentthatfisdefinedateveryrealnumberkanditsvalueatkisk5.

Itisalsoobservedthat,

Hence,fiscontinuousateveryrealnumberandtherefore,itisacontinuousfunction.

(b)Thegivenfunctionis

Foranyrealnumberk5,weobtain

Hence,fiscontinuousateverypointinthedomainoffandtherefore,itisacontinuousfunction.

(c)Thegivenfunctionis

Foranyrealnumberc5,weobtain

Hence,fiscontinuousateverypointinthedomainoffandtherefore,itisacontinuousfunction.

(d)Thegivenfunctionis

Thisfunctionfisdefinedatallpointsoftherealline.

Letcbeapointonarealline.Then,c5

CaseI:c

Therefore,fiscontinuousatallrealnumberslessthan5.

CaseII:c=5

Then,


CaseIII:c>5

Q4:Provethatthefunction iscontinuousatx=n,wherenisapositiveinteger.

Answer:

Thegivenfunctionisf(x)=x

Itisevidentthatfisdefinedatallpositiveintegers,n,anditsvalueatnisn .

Therefore,fiscontinuousatn,wherenisapositiveinteger.

Q5:Isthefunctionfdefinedby

continuousatx=0?Atx=1?Atx=2?

Answer:

n

n

Thegivenfunctionfis

Atx=0,

Itisevidentthatfisdefinedat0anditsvalueat0is0.


Atx=1,

fisdefinedat1anditsvalueat1is1.

Thelefthandlimitoffatx=1is,

Therighthandlimitoffatx=1is,

Therefore,fisnotcontinuousatx=1

Atx=2,

fisdefinedat2anditsvalueat2is5.


Q6:Findallpointsofdiscontinuityoff,wherefisdefinedby

Answer:

Thegivenfunctionfis

Itisevidentthatthegivenfunctionfisdefinedatallthepointsoftherealline.

Letcbeapointontherealline.Then,threecasesarise.

(i)c2

(iii)c=2

Case(i)c2

Case(iii)c=2

Then,thelefthandlimitoffatx=2is,


Itisobservedthattheleftandrighthandlimitoffatx=2donotcoincide.


Hence,x=2istheonlypointofdiscontinuityoff.


Answer:

Thegivenfunctionfis

Thegivenfunctionfisdefinedatallthepointsoftherealline.

Letcbeapointontherealline.

CaseI:

Therefore,fiscontinuousatallpointsx,suchthatx3



Answer:

Thegivenfunctionfis

Itisknownthat,

Therefore,thegivenfunctioncanberewrittenas



CaseI:

Therefore,fiscontinuousatallpointsx

CaseIII:

Therefore,fiscontinuousatallpointsx,suchthatx>0



Answer:

Thegivenfunctionfis

Itisknownthat,

Therefore,thegivenfunctioncanberewrittenas

Letcbeanyrealnumber.Then,

Also,

Therefore,thegivenfunctionisacontinuousfunction.

Hence,thegivenfunctionhasnopointofdiscontinuity.


Answer:

Thegivenfunctionfis



CaseI:


Hence,thegivenfunctionfhasnopointofdiscontinuity.


Answer:

Thegivenfunctionfis



CaseI:


Thus,thegivenfunctionfiscontinuousateverypointontherealline.

Hence,fhasnopointofdiscontinuity.


Answer:

Thegivenfunctionfis



CaseI:


Thus,fromtheaboveobservation,itcanbeconcludedthatx=1istheonlypointofdiscontinuityoff.

Q13:Isthefunctiondefinedby

acontinuousfunction?

Answer:

Thegivenfunctionis



CaseI:


Thus,fromtheaboveobservation,itcanbeconcludedthatx=1istheonlypointofdiscontinuityoff.

Q14:Discussthecontinuityofthefunctionf,wherefisdefinedby

Answer:

Thegivenfunctionis

Thegivenfunctionisdefinedatallpointsoftheinterval[0,10].

Letcbeapointintheinterval[0,10].

CaseI:

Therefore,fiscontinuousintheinterval[0,1).

CaseII:



Itisobservedthattheleftandrighthandlimitsoffatx=1donotcoincide.


CaseIII:

Therefore,fiscontinuousatallpointsoftheinterval(1,3).

CaseIV:





CaseV:

Therefore,fiscontinuousatallpointsoftheinterval(3,10].

Hence,fisnotcontinuousatx=1andx=3


Answer:

Thegivenfunctionis

Thegivenfunctionisdefinedatallpointsoftherealline.


CaseI:

Therefore,fiscontinuousatallpointsx,suchthatx





CaseV:


Hence,fisnotcontinuousonlyatx=1


Answer:

Thegivenfunctionfis

Thegivenfunctionisdefinedatallpointsoftherealline.


CaseI:

Therefore,fiscontinuousatallpointsx,suchthatx




CaseIII:

Therefore,fiscontinuousatallpointsoftheinterval(1,1).

CaseIV:




CaseV:


Thus,fromtheaboveobservations,itcanbeconcludedthatfiscontinuousatallpointsoftherealline.

Q17:Findtherelationshipbetweenaandbsothatthefunctionfdefinedby

iscontinuousatx=3.

Answer:

Thegivenfunctionfis

Iffiscontinuousatx=3,then

Therefore,from(1),weobtain

Therefore,therequiredrelationshipisgivenby,

Q18:Forwhatvalueof isthefunctiondefinedby

continuousatx=0?Whataboutcontinuityatx=1?

Answer:

Thegivenfunctionfis

Iffiscontinuousatx=0,then

Therefore,thereisnovalueofforwhichfiscontinuousatx=0

Atx=1,

f(1)=4x+1=41+1=5

Therefore,foranyvaluesof,fiscontinuousatx=1

Q19:Showthatthefunctiondefinedby isdiscontinuousatallintegralpoint.

Here denotesthegreatestintegerlessthanorequaltox.

Answer:

Thegivenfunctionis

Itisevidentthatgisdefinedatallintegralpoints.

Letnbeaninteger.

Then,

Thelefthandlimitoffatx=nis,

Therighthandlimitoffatx=nis,

Itisobservedthattheleftandrighthandlimitsoffatx=ndonotcoincide.

Therefore,fisnotcontinuousatx=n

Hence,gisdiscontinuousatallintegralpoints.

Q20:Isthefunctiondefinedby continuousatx= ?

Answer:

Thegivenfunctionis

Itisevidentthatfisdefinedatx= .

Therefore,thegivenfunctionfiscontinuousatx=

Q21:Discussthecontinuityofthefollowingfunctions.

(a)f(x)=sinx+cosx

(b)f(x)=sinxcosx

(c)f(x)=sinxxcosx

Answer:

Itisknownthatifgandharetwocontinuousfunctions,then

arealsocontinuous.

Ithastoprovedfirstthatg(x)=sinxandh(x)=cosxarecontinuousfunctions.

Letg(x)=sinx

Itisevidentthatg(x)=sinxisdefinedforeveryrealnumber.

Letcbearealnumber.Putx=c+h

Ifxc,thenh0

Therefore,gisacontinuousfunction.

Leth(x)=cosx

Itisevidentthath(x)=cosxisdefinedforeveryrealnumber.


Ifxc,thenh0

h(c)=cosc

Therefore,hisacontinuousfunction.

Therefore,itcanbeconcludedthat

(a)f(x)=g(x)+h(x)=sinx+cosxisacontinuousfunction

(b)f(x)=g(x)h(x)=sinxcosxisacontinuousfunction

(c)f(x)=g(x)h(x)=sinxcosxisacontinuousfunction

Q22:Discussthecontinuityofthecosine,cosecant,secantandcotangentfunctions,

Answer:

Itisknownthatifgandharetwocontinuousfunctions,then

Ithastobeprovedfirstthatg(x)=sinxandh(x)=cosxarecontinuousfunctions.

Letg(x)=sinx

Itisevidentthatg(x)=sinxisdefinedforeveryrealnumber.


Ifx c,thenh 0

Therefore,gisacontinuousfunction.

Leth(x)=cosx



Ifxc,thenh0

h(c)=cosc

Therefore,h(x)=cosxiscontinuousfunction.

Itcanbeconcludedthat,

Therefore,cosecantiscontinuousexceptatx=np,nZ

Therefore,secantiscontinuousexceptat

Therefore,cotangentiscontinuousexceptatx=np,nZ

Q23:Findthepointsofdiscontinuityoff,where

Answer:

Thegivenfunctionfis

Itisevidentthatfisdefinedatallpointsoftherealline.

Letcbearealnumber.

CaseI:


CaseIII:




Fromtheaboveobservations,itcanbeconcludedthatfiscontinuousatallpointsoftherealline.

Thus,fhasnopointofdiscontinuity.

Q24:Determineiffdefinedby

isacontinuousfunction?

Answer:

Thegivenfunctionfis


Letcbearealnumber.

CaseI:

Therefore,fiscontinuousatallpointsx0

CaseII:


Fromtheaboveobservations,itcanbeconcludedthatfiscontinuousateverypointoftherealline.

Thus,fisacontinuousfunction.

Q25:Examinethecontinuityoff,wherefisdefinedby

Answer:

Thegivenfunctionfis


Letcbearealnumber.

CaseI:


CaseII:


Fromtheaboveobservations,itcanbeconcludedthatfiscontinuousateverypointoftherealline.

Thus,fisacontinuousfunction.

Q26:Findthevaluesofksothatthefunctionfiscontinuousattheindicatedpoint.

Answer:

Thegivenfunctionfis

Thegivenfunctionfiscontinuousat ,iffisdefinedat andifthevalueofthefat

equalsthelimitoffat .

Itisevidentthatfisdefinedat and

Therefore,therequiredvalueofkis6.


Answer:

Thegivenfunctionis

Thegivenfunctionfiscontinuousatx=2,iffisdefinedatx=2andifthevalueoffatx=2equalsthelimitoffatx=2

Itisevidentthatfisdefinedatx=2and

Therefore,therequiredvalueof .


Answer:

Thegivenfunctionis

Thegivenfunctionfiscontinuousatx=p,iffisdefinedatx=pandifthevalueoffatx=pequalsthelimitoffatx=p

Itisevidentthatfisdefinedatx=pand

Therefore,therequiredvalueof


Answer:

Thegivenfunctionfis

Thegivenfunctionfiscontinuousatx=5,iffisdefinedatx=5andifthevalueoffatx=5equalsthelimitoffatx=5

Itisevidentthatfisdefinedatx=5and

Therefore,therequiredvalueof

Q30:Findthevaluesofaandbsuchthatthefunctiondefinedby

isacontinuousfunction.

Answer:

Thegivenfunctionfis

Itisevidentthatthegivenfunctionfisdefinedatallpointsoftherealline.

Iffisacontinuousfunction,thenfiscontinuousatallrealnumbers.

Inparticular,fiscontinuousatx=2andx=10

Sincefiscontinuousatx=2,weobtain

Sincefiscontinuousatx=10,weobtain

Onsubtractingequation(1)fromequation(2),weobtain

8a=16

a=2

Byputtinga=2inequation(1),weobtain

22+b=5

4+b=5

b=1

Therefore,thevaluesofaandbforwhichfisacontinuousfunctionare2and1respectively.

Q31:Showthatthefunctiondefinedbyf(x)=cos(x )isacontinuousfunction.

Answer:

Thegivenfunctionisf(x)=cos(x )

Thisfunctionfisdefinedforeveryrealnumberandfcanbewrittenasthecompositionoftwofunctionsas,

f=goh,whereg(x)=cosxandh(x)=x

Ithastobefirstprovedthatg(x)=cosxandh(x)=x arecontinuousfunctions.

Itisevidentthatgisdefinedforeveryrealnumber.

Letcbearealnumber.

Then,g(c)=cosc

2

2

2

2

Therefore,g(x)=cosxiscontinuousfunction.

h(x)=x

Clearly,hisdefinedforeveryrealnumber.

Letkbearealnumber,thenh(k)=k


Itisknownthatforrealvaluedfunctionsgandh,suchthat(goh)isdefinedatc,ifgiscontinuousatcandiffiscontinuousatg(c),then(fog)iscontinuousatc.

Therefore, isacontinuousfunction.

Q32:Showthatthefunctiondefinedby isacontinuousfunction.

Answer:

Thegivenfunctionis


f=goh,where

Ithastobefirstprovedthat arecontinuousfunctions.

2

2

Clearly,gisdefinedforallrealnumbers.

Letcbearealnumber.

CaseI:

Therefore,giscontinuousatallpointsx,suchthatx0

CaseIII:

Therefore,giscontinuousatx=0

Fromtheabovethreeobservations,itcanbeconcludedthatgiscontinuousatallpoints.

h(x)=cosx



Ifxc,thenh0

h(c)=cosc

Therefore,h(x)=cosxisacontinuousfunction.



Q33:Examinethat isacontinuousfunction.

Answer:


f=goh,where

Ithastobeprovedfirstthat arecontinuousfunctions.


Letcbearealnumber.

CaseI:

Therefore,giscontinuousatallpointsx,suchthatx

CaseII:

Therefore,giscontinuousatallpointsx,suchthatx>0

CaseIII:



h(x)=sinx

Itisevidentthath(x)=sinxisdefinedforeveryrealnumber.

Letcbearealnumber.Putx=c+k

Ifxc,thenk0

h(c)=sinc




Q34:Findallthepointsofdiscontinuityoffdefinedby .

Answer:

Thegivenfunctionis

Thetwofunctions,gandh,aredefinedas

Then,f=gh

Thecontinuityofgandhisexaminedfirst.


Letcbearealnumber.

CaseI:

Therefore,giscontinuousatallpointsx,suchthatx0

CaseIII:



Clearly,hisdefinedforeveryrealnumber.

Letcbearealnumber.

CaseI:

Therefore,hiscontinuousatallpointsx,suchthatx1

CaseIII:

Therefore,hiscontinuousatx=1

Fromtheabovethreeobservations,itcanbeconcludedthathiscontinuousatallpointsoftherealline.

5.1 math

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