Limits, Continuity and Differentiability

Board: JEE Main |Exam:JEEMain

JEEMainstudynotesforLimits,ContinuityandDifferentiabilitybyAjayYadav.

KeyConcepts

StandardLimits

lim(x→0)sinx/x=1.lim(x→0)(1+x)¹/x=e.lim(x→0)(e^x-1)/x=1.lim(x→∞)x)x/(1+1/x)^x/x)x=e.

L'Hopital'sRule

Iff(a)=g(a)=0or∞,lim(x→a)f(x)/g(x)=limf'(x)/g'(x).Applyrepeatedlyifneeded.

Continuity

fcontinuousataifLHL=RHL=f(a).Types:removable(hole),jump,infinitediscontinuity.

Differentiability

f'(a)=lim(h→0)[f(a+h)-f(a)]/h.Differentiable⇒continuous.f(x)=|x|notdifferentiableat0.

ChainRule

dy/dx=(dy/du)(du/dx).Parametric:dy/dx=(dy/dt)/(dx/dt).

ImportantFormulas

limsinx/x	1(asx→0)
lim(1+x)¹/x	e(asx→0)
lim(ex-1)/x	1(asx→0)
L’Hopital	limf/g=limf'/g'

SolvedExamples

Example1:Findlim(x→0)sin5x/x.

Solution:=5limsin5x/(5x)=5.

Example2:Findlim(x→0)(1-cosx)/x².

Solution:=lim2sin²(x/2)/x²=2(1/2)²=1/2.

Example3:Checkcontinuityoff(x)=|x|/xatx=0.

Solution:LHL=-1,RHL=1.Discontinuous.

PracticeQuestions

1. Findlim(x→0)(e²x-1)/x.
2. Findlim(x→2)(x²-4)/(x-2).
3. Checkdifferentiabilityoff(x)=|x-2| at x=2.
4. Find dy/dx if x=acosθ, y=asinθ.
5. Apply L’Hopital: limx→0 sinx/x³.

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