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1.Limits and Continuity

Apr 04, 2018

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    LIMITS AND CONTINUITY

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    FUNCTION

    Definition

    Afunctionf is a rule that assigns to each valuex in a setD a unique value denotedf(x).

    The setD is the domain of the function.

    The set of all values off(x) produced asx varies over

    domain is range.

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    FUNCTION

    Domain and range

    Domain Consisted all values ofx that is possible

    Range The set ofy values whenx varies over the domain

    Example

    Given

    2

    y x

    2y xDomain :

    Range :

    { : }D x x R

    { : 0, }R y y y R

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    FUNCTION

    Common restriction for function

    Type offunction

    Example Restriction Remarks

    Root

    Reciprocal

    log or ln

    ,

    2,4,

    n a

    n

    1

    a

    log ( )

    l n ( )

    a

    a

    0a

    0a

    0a

    The function willbecome complex

    number if .

    Any number dividedby zero is undefined.

    0a

    log or ln foris undefined.

    0a

    Example: Domain and range

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    CONCEPT OF LIMITS

    The limit of a function is concerned with the behavior of afunction as the independent variable approaches

    a given value.If the limit of functionf(x) asx approaches the point a isthe valueL, then it is denoted as

    lim ( )x a

    f x L

    This is not the exact definition of a limit. The definitiongiven above is more of a working definition.

    This definition helps us to get an idea of just what limitsare and what they can tell us about functions.

    .

    In simpler terms, the definition says that as x get closerand closer to a then f(x) must getting closer andcloser to L.

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Consider the following example

    Estimate the value of the following limit.2

    22

    4 12lim

    2x

    x x

    x x

    Choose values ofx that get closer to closer to 2 and

    substitute the values into . Doing this, gives

    the following table:

    2

    24 12

    2x x

    x x

    x 1.5 1.9 1.999 1.9999 1.99999 2 2.00001 2.0001 2.001 2.01 2.5

    f(x) 5.0 4.158 4.002 4.0002 4.00002 3.99999 3.9999 3.999 3.985 3.4

    Approach to 2 from the right sideApproach to 2 from the left side

    CONCEPT OF LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    The function is going to 4 asxapproaches 2, then

    2

    22

    4 12lim 4.2x

    x xx x

    Using tables of values to guess the value of limits is simplynot a good way to get the value of a limit but theycan help us get a better understanding of what limits are.

    CONCEPT OF LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    One-sided limits: Right side

    Only look for limit at right side of the point.

    ( )f x

    a

    L

    Right-handed

    x approaches a from

    right hand side.

    We write as .lim ( )x a

    f x L

    CONCEPT OF LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    One-sided limits : Left side

    Only look for limit at left side of the point.

    ( )f x

    a

    L

    Left-handed

    lim ( )x a

    f x L

    x approaches a from

    left hand side.

    We write as .

    CONCEPT OF LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Two-sided limits

    >> Requires value of limits from right and left sides.Two-sided limit off(x) is exist if

    lim ( ) lim ( )x a x a

    f x f x L

    Then,

    lim ( )x a

    f x L

    Example: One and Two-sided limits

    The limitlim ( )x a

    f x L

    is called two-sided limit.

    CONCEPT OF LIMITS

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    INFINITE LIMITS

    Infinite limits

    Infinite limits occur when and .lim ( )x a

    f x lim ( )x a f x

    All limits that have infinite limit does not exist.

    Example: Infinite limits

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

    Some basic limits

    limx a

    k k

    Letaandkis any real number

    limx a

    x a

    Consider first limits at a point which islimits off(x) whenx approaches any point a.

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

    We will discuss algebraic techniques for computing limitsof:-

    Polynomial functions

    Rational functions

    Radical functions, (function involve )

    Piecewise functions

    Absolute functions

    n

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Polynomial functions

    Letf(x) is polynomial function

    Then,

    lim ( ) ( )x a

    f x f a

    2

    0 1 2( )n

    nf x c c x c x c x

    Example: Polynomial function

    TECHNIQUES OF COMPUTING LIMITS

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    , we have to factor d(x) and n(x)(by common factor (xa) ) then cancel out

    (SIMPLIFIED THE FUNCTION).

    Rational functions

    Letf(x) is rational function, with d(x) and n(x) polynomial function

    ( )( )

    ( )

    n xf x

    d x

    Then,( )

    lim ( ) ( )

    ( )x a

    n af x f a

    d a

    ,provided that .( ) 0d a

    But how if ???( ) 0d a

    1 If ( ) 0n a lim ( )x af x

    does not exist.

    2 If( ) 0n a

    0

    lim ( ) 0x a f x

    Example: Rational function

    TECHNIQUES OF COMPUTING LIMITS

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    ForRationalize is:-

    ,provided that .

    Radical functions, (function involve )n

    Letf(x) is radical function, with e(x) and m(x) radical function

    ( )( )

    ( )

    m xf x

    e x

    ( )

    lim ( ) ( )x a

    m a

    f x e a ( ) 0e a

    But how if ???( ) 0e a

    1 If

    2 If

    ( ) 0m a lim ( )x a

    f x

    does not exist.

    , rationalizee (x) or m (x)

    ( ) 0m a 0lim ( )0x a

    f x

    a b

    2

    2

    2

    ( )( )a b a b

    a b a b a b

    a b

    Example: Radical function

    TECHNIQUES OF COMPUTING LIMITS

    Then,Rationalize???

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Piecewise functions

    Piecewise function is defined by

    1

    2

    ,

    ,

    f x a x b

    f x f x b x c

    The limits of piecewise function is obtained by finding theone-sided limits first.

    Example: Piecewise function

    TECHNIQUES OF COMPUTING LIMITS

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Absolute functions

    Absolute function is defined by ( )f x b x

    The limits of absolute function is

    lim ( ) ( ) ( )x a

    f x f a b a

    But how if (A is any number)0( ) or ???0 0Ab a

    1 If lim ( )x af x

    does not exist.( )

    0

    Ab a

    TECHNIQUES OF COMPUTING LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    20

    ( )0

    b a If

    Example:Absolute function

    Consider these notes (a is any real number)

    ( )

    ( ) ( )

    x a x a

    f x x a x a x a

    ( )( )

    ( )

    a x x af x a x

    a x x a

    1

    2

    And always check limits from both sides!

    The limit is obtained by finding the one-sided limit first

    TECHNIQUES OF COMPUTING LIMITS

    ( )

    x x a

    f x x x x a

    ( )x x a

    f x xx x a

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

    Some basic limits

    limx

    k k

    Letkis any real number

    limx

    x

    Limits off(x) whenxincrease without bound ( )orxdecrease without bound ( ).

    x x

    limx

    x

    1lim 0x x

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    TECHNIQUES OF COMPUTING LIMITS

    We will discuss algebraic techniques for computing limitsof:-

    Polynomial functions

    Rational functions

    Radical functions, (function involve )n

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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    depends on the highest degree term and

    consider

    Polynomial functions

    lim ( )x

    f x

    Example: Polynomial function

    TECHNIQUES OF COMPUTING LIMITS

    Letf(x) is polynomial function

    Then,

    2

    0 1 2( )n

    nf x c c x c x c x

    , 1,3,5, ( )lim , lim

    , 2, 4, 6, ( )

    n n

    x x

    n odd x x

    n even

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Letf(x) is rational function( )( )

    ( )

    n xf x

    d x

    Then,

    lim ( )x

    f x

    Step 1 Divide each term in n(x) and d(x) with

    the highest power ofx.

    Step 2 Substitute to find the answer

    Example: Rational function

    Rational functions

    TECHNIQUES OF COMPUTING LIMITS

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    Letf(x) is radical function( )

    ( )( )

    m xf x

    e x

    Case 1 Case 2

    f(x) = m (x) only

    Step 1 Divide m(x) and e(x) with the highest

    power ofxStep 2 Substitute to find the answerx

    Rationalize m (x)

    Example: Radical function

    Radical functions, (function involve )

    TECHNIQUES OF COMPUTING LIMITS

    n

    Try substitute the value of into the function. If cantfind the answer, consider these cases:

    ( )( )

    ( )

    m xf x

    e x

    Special case? Use concept2x x

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    CONTINUITY

    We perceive the path of moving object as an unbrokencurve without gaps, breaks or holes. In this section,

    we translate the unbroken curve into a precisemathematical formulation called continuity.

    A functionf(x) is said to be continuous atx = a if

    lim ( ) ( )x a

    f x f a

    Example: Continuity

    Provided that EXIST and DEFINED.lim ( )x a

    f x

    ( )f a

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc
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    DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

    CONTINUITY

    Type of discontinuity

    Removable discontinuity

    Hole

    Jump discontinuity

    Jump

    00lim ( ) ( )

    x xf x f x

    0 0

    lim ( ) lim ( )x x x x

    f x f x

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    DEPARTMENT OF MATHEMATICS AND STATISTICS FSTPi

    Infinite discontinuity

    CONTINUITY

    Infinite

    Example: Discontinuity

    0 0lim ( ) or lim ( )x x x xf x f x

    http://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_6/1.Example%20Limits%20and%20continuity.doc