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# One Sided Limits calculus

Jul 08, 2018

## Documents

• 8/19/2019 One Sided Limits calculus

1/19

One-sided Limits

School of Electrical Engineering &Computer Sciences

• 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

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

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

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

• 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 =

• 8/19/2019 One Sided Limits calculus

7/19

x + 52

Example: Find the limit

x x x   −−→ 23

• 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 =

• 8/19/2019 One Sided Limits calculus

9/19

2)3(   ++   x x

Example: Find the limit

22   ++−→   x m

x

• 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

• 8/19/2019 One Sided Limits calculus

11/19

2)3(   ++   x x

Example: Find the limit

22   +−−→   x m

x

• 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

• 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

)( )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−

• 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

→ →

)( )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

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

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