Limits Section 2.1

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Sections 2.1 & 2.2

Fall 2010 Haitham S. Solh


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Calculus

y Algebra: Deals with static situations (finding a quantity)Calculus: Deals with dynamic situations (finding the rateat which a quantity is changing). It is basically the studyof change, just like geometry is the study of shape and

algebra is the study of operations and solving equations.y Fundamental notions include limits, continuity,

derivatives, integrals, and infinite series.y Calculus has two main branches: Differential Calculus and

Integral Calculus.y Calculus has widespread applications in science,

economics, and engineering and can solve many problemsfor which algebra alone is insufficient.



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Limitsy Without limits, calculus would not exist (Lial, p.53)y The idea of a limit in Mathematics resembles the meaning

of the word in English. It is a fundamental part inestablishing the notions of continuity and differentiability.

y Lets start with an intuitive approach: Letfbe a functionand c a real number.fis defined for all numbersx nearc,

but not necessarily at c itself. IfL is a real number, thenwe can say that

yL is the limit off(x) asx tends to c if and only iff(x) isclose toL for allx values which are close to c.

y Asx approaches c,f(x) approachesL


Lxfcx

!p

)(lim


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Limits: Examplesy

Example 1: Letf(x) = 4x + 5 and c = 2. Asx approaches 2,4x approaches 8, and 4x +5 approaches 8+5=13. Therefore,limf(x) = 13.

x2

y Example 2: Letf(x)=1 x and c= 8 . Asx approaches 81x approaches 9, and 1 x approaches 3. Therefore,limf(x) = 3.

x8

y Challenge: What is the limit off(x) asx approaches 2 inexample 2?

y limf(x) does not exist.

x

2Fall 2010 Haitham S. Solh


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Limits: A GraphicalAnalysisy

The graph of the functionfshows that asx approachesc along thex-axis, f(x)approachesy along the y-axis.

y As stated earlier,f(x) may ormay not be defined at c, whichmeans thatf(c) may ormay notexist.

y How manypossibilitiesexist?

y Going back to the first two examples, it is clear that thelimit in those cases was basically found by substituting thevalue ofc in the expression of the function.

Hence: ])([)()(lim existscfWhencfxfcx!

p


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Limits: Three PossibleCases

y As you can see, we have 3 possible cases:



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Limits: Three PossibleCasesy Example:

yNotice thatf(3) does not exist, sofis not defined atx = 3.However, remember that were trying to find the limit asx

approaches 3, not atx = 3. Thus, for allx near 3,f(x) can bere-written as [(x 3) (x + 3) /(x 3)] = x + 3. Hence,

yNow try the following examples:


39)(

2

3

!

p xxxfLimFind

x

633

9)(

3

2

3

!!

!

pp

xLix

xxfLi

xx


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One-SidedLimits



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One-SidedLimitsy Example:



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One-SidedLimitsy Example 2:


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Limits Section 2.1