Pengertian limit secara intuisi

  − −

  =

  ! " # # $ $$ $$$ $$$$ % $ $$

  $$$ $$$$ & & & &

  ' & " ( & ! )

  ) = − = −

  − →

  " * & − −

• ( (

  , =

  →

  = ∈ =

  → →

  ∈

  → →

  ± = ±

  { } → → →

  =

  → →

  =

  → → → →

  = ≠

  !

  → → →

  " #

  $ % %

  $ &

  ' ( − )

  # − *

  → →

• −

  →

  )

• →−

  − ) = − = − = ) + ) →

  − = − + = − , + = +

• !

  → → # − = − = = * * * *

  → − + − + + = = = −

• →−

  

− + − +

) ) )

  & ' %

  →

  − −

  →

  − + −

  ! !

  →

  − − − = = =

  − − − "

  #

• −

  → →

  − −

  = −

  →

  = + = + =

• !

  − − +

  =

  →

  − −

  → →

  − − +

  = −

  → →

  " #

  − − −

  − + − + − + = = =

  →

  !

• −
• −

  = =

Hitunglah nilai limit berikut ini!(Perasionalan Akar

• &mi
• − a.

  b.

  →− →− → →

  − −

  − −

  Solusi:

• − + − −

  ! a. = = =

  →

  − − − !

• − + − + + = ⋅

  → → − −

  − −

  ( )

  = =

  → →

  − − + + + +

  ( ( ) )

  − − + + + +

  ( ( ) ) ( ( ) )

  ( ( ) )

  − =

  = →

  − + +

  ( ) → ( )

  = = = =

  − + −

• −

  − ! = = = b.

  →− − − − − !

• − −

  = ⋅

  →− →−

  − −

• −

  −

  ( ( ) ) ( ( ) )

  = =

  →− →−

  − + + − + +

  ( ) ( ) )

  ) ( (

  − =

  =

  →− →−

• −

  ( ) ( )

  = = = =

• − +

• !
• !

  − →

• !
• →

  = = ⇔ =

• _{+ −}

  → → →

• ₊

  12

  → ≠

  − →

.

  →

  ≤ − + 

• 

  = − < < - $ .

   − + − ≥

• 

  →− →

  % % / /

• /

  #

  →− _{− −} + = = − + = →− →−

  = = − = _{+ +}

  →− →− _{− +} = = ⇒ = →− →− →−

  % /

  / − + # #

  → → ≤ − + 

• 

  = = = − −

  = − < < -  →− →

  − + − ≥

• 

  = − + = − + =

  → → ≠ ⇒

  − + →− → →

   ≤

   = < <

  " 

  

• ≥ 

  / →

• /

  → → /

  →

  1 ₂

  _{− −} f x x _{x → x →} lim ( ) = lim = =

  → f x x lim ( ) lim _{x → x →} = = _{+ +}

  1

  1 ¹ lim ( ) lim

  _{− −} 1 _{x x} f x x → →

  = = ₂ lim ( ) lim 2 3 f x x = + =

• ^{+ −}

  → → ≠

  1

• _{+ +} 3 _{x x} f x x

  2

  2

  2 lim ( ) lim2

  

6

x x f x x

  → → = + = ^{2 1 1} lim ( ) lim 2

  → → = + =

  →

  1 & &

  − − , − ! a. f.

  → ) →−

  −

• b.

  − →− g.

• →

  − c.

  → → ! ! − −

• h.
• &mi
• )

  → − d.

  → −

  −

• i.
• −

  e.

  →

  →− − )

  ,

  2 ;

  1  x x

≤

f x ( ) lim ( ) f x "

  =  _{x 1} →

  1 x

  1

>

 !

  2 ₂  x ; x ≤ 

  ( )

  1 & " f x = x < ≤ x   ₂ 

  1

  1  + x x >  lim ( ) f x lim ( ) f x _{x x 1} !

  2 → → x 2; x

  1

− − < −

 ₂  f x ( ) x ; 1 x

  1 3 " = − − ≤ <  ₂ 1 ;

  1 x x

   + ≥  lim ( ) f x lim ( ) f x _{x 1 x 1} !

  2 →− →

  2

  2

  !

  f x →

  lim ( ) _x

  3

  f x →

  − >  ¹ lim ( ) _x

  = < ≤  

  x x f x x x x

  3

  3 1,

  1 ( ) 5 ,1

  !

  4 " ₂ 3 2 ,

  f x →

  lim ( ) _x

  3

  f x →

  − >  ¹ lim ( ) _x

  = < ≤  

   + ≤ 

  x x f x x x x

  3

  3 3 1,

  1 ( ) 5, 1

• ≤ 5 "
₂
• ≤  

  → − + −

  → − +

• − _→ −

  →− − − −

  − −

  → − +

• − −

  − →

• =

  →

  − =

  − + − +

  →

  − + =

  − − −

  −

  x 

,

  1 x ≤ −

   + x

  1 

• ₊ ( ) ,-1

  1

!" f x = x < ≤ x #

_→   − 1 x , x >

  1  

  $ % x x

   

, ,

  1 x x ≤ − ≤ −

  1

   + x

  1  ( ) ,-1

  1

!" f x = x < ≤ x #

_{→ −}   − 1 x , x >

  1  

  $ %