Top Banner

of 8

One Sided Limits calculus

Jul 08, 2018

ReportDownload

Documents

  • 8/19/2019 One Sided Limits calculus

    1/19

    One-sided Limits

    Muhammad Nadeem

    School of Electrical Engineering &Computer Sciences 

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    2/19

    One-sided Limit  Let   ƒ( x) is defined on an interval (a, b),

    where a < b and   ƒ( x) approaches

    arbitrarily close to K as   x approaches a from within that interval, then ƒ has right- hand limit K at a. We write

    K  x f  Lim   =)(  x

     y

    b

    a x

    [email protected] 

    a x→

     M  x f  Lim b x

    = −

    )(

     Let   ƒ( x) is defined on an interval (a, b), where a < b and   ƒ( x) approaches arbitrarily close to M as   x approaches b

    from within that interval, then ƒ has Left- hand limit M at b. We write

     x

     y

    ba x

  • 8/19/2019 One Sided Limits calculus

    3/19

    Limit  Let  ƒ ( x) is defined on an interval (a, b), where a < b and c is any

    number within that interval. Function   ƒ ( x) has a limit as   x

    approaches c if and only if it has left-hand and right-hand limits at x=c and these one-sided limits are equal:

     L x Lim   =)(   y

    [email protected] 

    c x   +

     L x f  Lim c x

    = −

    )(

     L x f  Lim c x

    = →

    )(  xb

     L

    a x   c x

  • 8/19/2019 One Sided Limits calculus

    4/19

     y

    A function may fail to have a Limit at a point in its domain

    1

    

     

    =

    ≠ =

    0 0

    0  1

    )(

     x

     x  x xg

     y

      

    < =

    1 1

    1 0 )(

     x

     x  xU 

     x 0 0

     x

    [email protected] 

    1)( 0

    = +

     x f  Lim  x

    0)( 0

    = −

     x f  Lim  x

    ∞= +

    )( 0

     x f  Lim  x

    −∞= −

    )( 0

     x f  Lim  x

  • 8/19/2019 One Sided Limits calculus

    5/19

     x + 52

    Example: Find the limit

     x x x   −+→ 23

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    6/19

    5)3(2

    )(

    )52( 52

    2

    2

    3

    3

    23

    + =

    +

    =

    +

    +

    +

    +

    →   x x Lim

     x Lim

     x x

     x  Lim

     x

     x

     x

    6

    11 =

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    7/19

     x + 52

    Example: Find the limit

     x x x   −−→ 23

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    8/19

    5)3(2

    )(

    )52( 52

    2

    2

    3

    3

    23

    + =

    +

    =

    +

    →   x x Lim

     x Lim

     x x

     x  Lim

     x

     x

     x

    6

    11 =

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    9/19

    2)3(   ++   x x

    Example: Find the limit

    22   ++−→   x m

     x

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    10/19

    )2)(3(2)3(   ++ =

    ++

    ++

     x x  Lim

     x x  Lim

     

     

    −+

    =+

    2 )2(

    2 0

    2 )2(

    2

     x x

     x

     x x

     xSince

    1

    )3( 2

    +=

    += +

    −→

    −−

     x Lim  x

     x

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    11/19

    2)3(   ++   x x

    Example: Find the limit

    22   +−−→   x m

     x

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    12/19

    )2)(3(2)3(   ++− =

    ++

    −−

     x x  Lim

     x x  Lim

     

     

    −+

    =+

    2 )2(

    2 0

    2 )2(

    2

     x x

     x

     x x

     xSince

    1

    )3( 2

    −=

    +−= −

    −→

    −−

     x Lim  x

     x x

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    13/19

      

     

    ≤<

  • 8/19/2019 One Sided Limits calculus

    14/19

    2)()( )1( 2

    11 =+= −− →→  x x Lim x f  Lim  x x

    222)( )2( 11

    =+= ++

    →→

     x Lim x f  Lim  x x

    2)( )3( 1

    = →

     x f  Lim  x

     

    Since LHL=RHL=2

    [email protected] 

    )( )6( 2

     x f  Lim  x→

      22

    == −− →→   x x

    4

    1

    2

    1  

    4

    2 )( )5(

    2 2

    22

    − =

    +

    − =

    − =

    +++ →→→   x  Lim

     x

     x  Lim x f  Lim

     x x x

    Doesn't exists

  • 8/19/2019 One Sided Limits calculus

    15/19

      

     

    ≤<

  • 8/19/2019 One Sided Limits calculus

    16/19

     x

     y

    1

    0

    2

    3

    1 2 3 41−2−3−

    1−

    2−

    3−

    [email protected] 

  • 8/19/2019 One Sided Limits calculus

    17/19

    2)( )1( 2

    −= +

    −→

     x f  Lim  x

    0)( )2( 1

    = −

    −→  x f  Lim

     x

    1)( )3( 1

    = +

    −→

     x f  Lim  x

    → −→

    )( )4( 1

     x f  Lim  x

    0)( )5( 0

    = −

     x f  Lim  x

    3)( )6( 0

    = +

     x f  Lim  x

    Doesn't exists

    [email protected] 

    → →

    )( )7( 0

     x f  Lim  x

    1)( )8( 2

    = −

     x f  Lim  x

    1)( )9( 2

    = +

     x f  Lim  x

    1)( )10(

    2

    = →

     x f  Lim

     x

    2)( )11( 4

    = −

     x f  Lim  x

    Doesn't exists

  • 8/19/2019 One Sided Limits calculus

    18/19

    Example: Let f is an odd function of x and

    3)( 0

    = +

     x f  Lim  x

    [email protected] 

    Can you guess

    If yes, write limit, if no, give reason.

    ?)( 0

    = −

     x f  Lim  x

  • 8/19/2019 One Sided Limits calculus

    19/19

    Since  f ( x) is odd function, that is

    )()(   x f  x f    −=−

    and

    Hence

    3)( 0

    = +

     x f  Lim  x

    3)( 0

    −= −

    →  x f  Lim

     x

    [email protected]