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JEECONCEPTSBOOSTER
LIMITSANDDERIVATIVES
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
1.INTRODUCTION
1.Definelimitwhatitrepresents
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
2.CONCEPTOFLIMITS
1.conditionofexistenceoflimitevaluationofrighthandlefthandlimit
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
2.CONCEPTOFLIMITS
2.Relationbetweenthevalueofafunctionatapointandthelimitatapoint
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
2.CONCEPTOFLIMITS
3.Limits:FewExamples
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
3.ALGEBRAOFLIMITS
1.Thealgebraoflimits
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
4.INDETERMINATEFORM
1.Introductionofallindeterminateforms
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
1.(i)Directsubstitution
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
2.(ii)Factorisationmethod(0/0form)
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
3.Limits:Exampleoffactorisationmethod
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
4.(iii)Rationalisationmethod(0/0or∞/∞form):
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
5.Limits:ExampleofRationalisation
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
6.(iv)Byusingsomestandardlimits.
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
135.EVALUATIONOFALGEBRAICLIMITS
7.ExamplesonStandardlimits.
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
8.(v)methodofevaluationofalgebraiclimitsatinfinity
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
5.EVALUATIONOFALGEBRAICLIMITS
9.ExamplesonEvaluationOfAlgebraicLimits
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
6.EVALUATIONOFTRIGONOMETRICLIMITS
1.BasicTrigonometrictheoremforlimits
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
6.EVALUATIONOFTRIGONOMETRICLIMITS
2.Evaluationoftrigonometriclimitwhenvariabletendstononzeroterm(algorithm)
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
6.EVALUATIONOFTRIGONOMETRICLIMITS
3.ExamplesbasedonEvaluationoftrigonometriclimits.
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
7.USEOFEXPANSIONSINEVALUATINGLIMITS
1.Allimportantexpansions
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
7.USEOFEXPANSIONSINEVALUATINGLIMITS
2.Provethat(i)
limx → 0
= loge a
(ii) limx → 0
= 1
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
7.USEOFEXPANSIONSINEVALUATINGLIMITS
3.Evaluationoflimitoftheform1 ∞
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
ax − 1x
log1+ x
x
227.USEOFEXPANSIONSINEVALUATINGLIMITS
4.Calculationoflimitsbytakinglogarithm
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
8.LHOPITALRULEFOREVALUATINGLIMITS
1.Howtoapplyconditionon:L'Hopital'RuleForEvaluatingLimits
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
8.LHOPITALRULEFOREVALUATINGLIMITS
2.Findingunknownwhenlimitisgiven
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
9.GEOMETRICALMEANINGOFADERIVATIVE
1.DefinitionofGeometricalmeaningofDerivativeanditsExamples
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
9.GEOMETRICALMEANINGOFADERIVATIVE
2.Physicalinterpretationofderivativeofapoint
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
9.GEOMETRICALMEANINGOFADERIVATIVE
3.Geometricinterpretationofderivativeatapoint
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
9.GEOMETRICALMEANINGOFADERIVATIVE
4.Derivativeofafunction(i)asaratemeasurer
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
1. (i)Iff(x) = xn ; where nε R then the differentiation ofxn with respect to x isnxn − 1
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
2. (ii)The differentiation of ex with respect to x is ex. (iii) The differentiatiion ofax(a > 0; a ≠ 1)withrespecttoxisax loge a
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
1
31 3. (iv)The differentiation of loge x; x > 0is (v) The differentiation of
loga x(a > 0; a ≠ 1)withrespecttoxis
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
4.(vi)Thedifferentiationofsinxwithrespecttoxiscosx
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
5.(vii)Thedifferentiationofcosxwithrespecttoxis-sinx
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
6.(viii)Thedifferentiationoftanxwithrespecttoxissec2 x
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
7.(ix)Thedifferentiationofcotxwithrespecttoxis−cos ec2x
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1x
1x loge a
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
8.(x)Thedifferentiationofsecxwithrespecttoxissec x tan x
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
9.(xi)Thedifferentiationofcosecxwithrespecttoxis-cosecxcotx
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
10.Someotherderivationbyfirstprincipal:
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
10.THEOREMSONDERIVATIVES
11.Examplebasedontheoremofderivatives
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
1.Fundamentalrulesofdifferentiation
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
2.Fundamentalrulesofdifferentiation-Subtraction
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
3.Fundamentalrulesofdifferentiation-Multiplication
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
4.Fundamentalrulesofdifferentiation-DIVISION
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
5.Theorem:(i)Differentiationofaconstantfunctionis0(ii)Letf(x)bethedifferentiablefunctionandletcbeaconstant.Thencf(x)isalsodifferentiablesuchthat
d = c
. d( )
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
6.Differentiatethefollowingfunctionwithrespecttox(i)logx x(ii)e3 log x
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
11.FUNDAMENTALRULESOFDIFFERENTIATION
7. If f(x) andg(x) aredifferentiable function ; then show thatf(x) ± g(x) are alsodifferentiablesuchthat
d
= d ± d
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CONCEPTFORJEE||ChapterLIMITSANDDERIVATIVES
12.SandwichTheorem
cf(x)dx
f(x)
dx
f(x) ± g(x)dx
f(x)
dx
g(x)
dx
471.LimitscalculationbySandwichTheorem
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