8/12/2019 45870198-limitet

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Ushtrime t zgjidhura

Limitet dhe zbatimi i limiteve

Ass.Halil Snope

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LIMITT-T gjinden limitet e mposhtme:

) ( ) ( )

) ( ) ( ) ( ) ( )

)

) ( )

)

2 2

2 2

2 2

3 3

2 2

5

2 2

5 5

1 lim 3 5 2 lim 3 2 5 2 2 4

2 lim 1 1 lim 3 1 3 1 16

1 2 1 33 lim lim

2 2 2 4

34 lim 0 0 lim

5

3 10 5 2 105 lim lim l

5 5

x x

x x

x x

x

x x

x x

x x

x

x

xpasi emruesi tenton n dhe numruesi iti nuk ekziston

x

x x x x x

x x

+ = + =

+ = + =

+ += =

+ ++

+ = =

( ) ( )

( ) ( )( )

) ( ) ( )

( ) ( )( )

) ( ) ( )

( )

5

5 5

2 2

2 2 2

2 2

2 2

2 22 2 2

5 2 5im

5

2 5lim lim 2 7

5

2 3 26 2 3 66 lim lim lim

2 2 2

3 2lim lim 3 5

2

3 2 36 3 2 67 lim lim lim

23 2 2 2

x

x x

x x x

x x

x x x

x x x

x

x xx

x

x x xx x x x x

x x x

x xx

x

x x xx x x x x

x xx x x x x

+ =

+

= = + =

+ + + = = = =

+

= = + =

+ + = =

++ + + + +

( ) ( )

( ) ( )

)( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )

) ( )

( )( )

2 2

2 2

21 1 1

1 1

22

4 4 4

( 2)

2 3 3 5lim lim 5

1 2 1 1

1 5 14 5 5 58 lim lim lim

1 1 1 11

5 1 5 6lim lim 3

1 1 1 2

22 2 29 lim lim lim

4 4 2 4 2

x x

x x x

x x

x x x

x

x x x

x x x

x x xx x x x x

x x x xx

x x x

x x x

xx x x

x x x x x

=+ +

+ = = = =

+ + +

+ + + = = = + +

+ += = = =

+ +

+= = =

+ +

( )( )

) ( )

( )( )

( )( )

4 4

22

9 9 9

9 9

4 1 1lim lim

424 2

33 3 310 lim lim lim

9 9 3 9 3

9 1 1lim lim

639 3

x x

x x x

x x

x

xx x

xx x x

x x x x x

x

xx x

= = =

+ + +

= = = + +

= = =

+ +

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) ( )

( )( )

( )( ) ( )( )

) ( )

( )( )

22

3 3 3

3 3 3

22

5 5 5

5

1 21 2 1 2 1 211 lim lim lim

3 3 1 2 3 1 2

1 4 3 1 1lim lim lim

41 23 1 2 3 1 2

2 1 32 1 3 2 1 3 2 1 312 lim lim lim

5 5 2 1 3 5 2 1 3

2lim

x x x

x x x

x x x

x

xx x x

x x x x x

x x

xx x x x

xx x x

x x x x x

x

+ + + + += = =

+ + + +

+ = = = =

+ + + + + +

+= = =

+ +

=

( )( )( )

( )( )

) ( )

5 5

5

0

2 51 9 2lim lim

2 1 35 2 1 3 5 2 1 3

2 1lim 6 3

13 lim 2sin cos 2sin 0 cos0 2 0 1 1

x x

x

x

x

xx x x x

x x

= = =

+ + +

= =

+ = + = + =o o

)14 T c!tohet limiti i djtht dhe i mjt3 3

lim ( ) lim ( )x x

f x dhe f x +

22 " 3( )

3 " 3

x x pr xnse f x

x pr x