Μαθηματικά θετικού προσανατολισμού ΟΡΙΑ

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Ορια Μαθηματικά Γ’ Λυκείου mathhmagic.blogspot.gr ΕΠΙΜΕΛΕΙΑ : Ο ΑΛΙ ΜΠΑΜΠΑ ΚΑΙ ΟΙ ΣΑΡΑΝΤΑ ΚΛΕΦΤΕΣ………………. 53 ΜΑΘΗΜΑΤΙΚΑ Γ ΛΥΚΕΙΟΥ ΟΡΙΑ ΟΜΑΔΑ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ ΘΕΤΙΚΩΝ ΣΠΟΥΔΩΝ, ΟΙΚΟΝΟΜΙΑΣ ΚΑΙ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΙΜΕΛΕΙΑ: Μήταλας Γ , Δρούγας Α. Χάδος Χ. Γερμανός Ξ. Πάτσης Σ. Ο ΤΣΕΛΕΜΕΝΤΕΣ ΤΟΥ ΥΠΟΨΗΦΙΟΥ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ ΘΕΤΙΚΟΥ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ
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  1. 1. mathhmagic.blogspot.gr : . 53 , : , . . . .
  2. 2. mathhmagic.blogspot.gr : .54 . , , , . , ., viral, on line . , . . . . !! .... 1 4 1/2 1/2 1 2 1/2 1/2 1 1 1 2 4 6 1 2 1 . . . . . , . . . . . . . . . , . . . . . .
  3. 3. mathhmagic.blogspot.gr : . 55 1) , , ,. ... 2) , ., 3) , . , 4) , , 5) , ., 6) , ., 7) , ., 8) , . , 9) , , 10) ,. . , 11)- . , 12)- .. , 13)- .., 14) 1,2 , ., 15) 1000+1 , ., 16) , & ., 17) , .... 18) 19), . 20) ,., 21) ,-, 22) .. . 23),., 24), . 25) , Spivak M. , . 26) . , 24)Problems in Calculus ,..Maron Mir Publisher 25) , . , 26) 27) , , ( qr-code) .
  4. 4. mathhmagic.blogspot.gr : .56 3 9 )( 2 = x x xf . f {3}, x = 3 3 3 =0. x, 3. x 2.9 2.99 3.001 3.01 3.1 f(x) 5.9 5.99 6.001 6.01 6.1 , x () 3, , f(x) 6. x 3, , f(x) 6. : f(x) 6 , x 3. : 3x , 6)( xf 3 lim ( ) 6 x f x = f (, x0) (x0, ). f(x) l, x x0 ( x0), l x x0. 0 lim ( ) x x f x l ==== A f(x) x0, . f. , x = x0, . x, x0 f(x). , x x, f(x) l. , ( ) 0x x lim f x 0 x , : 1). f x ( ) 0 x . 2). 0y ( )f x ( ) = 0 0 0 x x x ,y lim f x . 21 ( ) 0x x lim f x
  5. 5. mathhmagic.blogspot.gr : . 57 ( ) ( ) ( ) 0 0 0x x x x x x lim f x lim f x lim f x + = = = . 0x , ( ) 0x . f ( )0x , , ( )0,x ( ) ( ) 0 0x x x x lim f x lim f x+ = . f ( )0,x , ( )0x , , ( ) ( ) 0 0x x x x lim f x lim f x = . 21-1. f 0x 0,1,2,3,4= , f : x 0 : x 0 ( ) x 0 lim f x 1 = x 0+ ( ) x 0 lim f x 1+ = . , 0x 0= ( )x 0 limf x 1 = . x 1 : x 1 ( ) x 1 lim f x 1 = x 1+ ( ) x 1 lim f x 0,5 1+ = . , f 0x 1= . x 2 : x 2 ( ) x 2 lim f x = x 2+ ( ) x 2 lim f x+ = . f 0x 2= ( )x 2 limf x = . x 3 : x 3 ( ) x 3 lim f x = + x 3+ ( ) x 3 lim f x 0,5+ = . f 0x 3= . x 4 : x 4 ( ) x 4 lim f x 1 = x 4+ ( ) x 4 lim f x 1+ = . f 0x 4= ( )x 4 limf x 1 = .
  6. 6. mathhmagic.blogspot.gr : .58 21-1. f . i) ( )x 2 lim f x ii) ( ) x 1 lim f x iii) ( )+ x 1 lim f x iv) ( )x 1 lim f x v) ( )x 2 lim f x =0 x 1; i) x 2 ( ) = x 2 lim f x 1. ii) x 1 : x 1 ( ) = x 1 lim f x 1 iii) + x 1 ( )+ = x 1 lim f x 1 1 . , f 0x 1= . iv) x 1 ( ) = x 1 limf x 1 v) x 2 ( ) = + x 2 lim f x vi) + x 2 ( )+ = x 2 lim f x 1 . f =0 x 2 . Cf
  7. 7. mathhmagic.blogspot.gr : . 59 21-2. 0x x lim f(x) 0f(x ) , , f : 21-3. f [-2,+ ) . . i) 2)(lim 2 = xf x ii) 1)(lim 1 = + xf x iii) 2)(lim 1 = xf x iv) 3)(lim 2 = xf x v) 4)(lim 3 = xf x vi) 3)(lim 4 = xf x 21-4. f , , )(lim 0 xf xx , : i) 2 x -5x+6 f(x)= x-2 , 0x =2 ii) x, x 1 f(x)= 1 , x>1 x , 0x =1 iii) 2 x , x 1 f(x)= -x+1, x>1 , 0x =1 iv) 2 x f(x)=x+ x , 0x =0 . 21-5. f , , )(lim 0 xf xx , : i) 3 2 2 x +3x -x-3 f(x)= x -1 , 0x =1 0x = -1 ii) 2 (x+1) 9x -6x+1 f(x)= 3x-1 , 0 1 x = 3 . 21-6. ( ) 2 2x x -2x+1 f x = 1-x . fC , , ( ) ( ) ( ) ( )x 1x 1 x 1 1 ,lim f x ,lim f x , lim f x+ f . 21-7. ( ) x f x = x 0x ( ) 0x x lim f x . O x0=3 y=f(x) 3 2 x y O y=f(x) x0=2 2 4 2 x y O x0=1,2 y=f(x) 1 1 2 2 x y O x0=1,2,3 y=f(x) 1 1 2 32 x y O y=f(x) 1 1 4 3 2 43-2 2 x y
  8. 8. mathhmagic.blogspot.gr : .60 22 1. R f g 0x : ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + = = = = = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 1). lim f x g x lim f x lim g x 2). lim f x lim f x 3). lim f x g x lim f x lim g x lim f xf x 4). lim lim g x 0 g x lim g x 5). lim f x lim f x 6). lim f x lim f x f( ) ( ) ( )( ) = 0 0 0 x x x x x 0 x 7). lim f x lim f x , * : ( ) ( )( )0x x lim f x g x + f, g 0x . = = = 0 0 0 0 0x x x x x x lim x x , lim C C , lim x x * . ( ) ( )P x ,Q x , ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 x x x x x x x x 0 P xP x lim P x P x lim = Q x 0 lim x=x lim x=x Q x Q x = ( ) 0x x lim f x = , . ( ) ( )( ) ( ) ( ) ( ) 0 0 0 0 0x x x x x x x x x x lim f x lim f x 0 lim f x 0 lim f x lim f x + = = = = = . (1) (7); . , , , : 1) . 2) , , , , 1: x 1 2 x 2 3 x 3 lim x 2 2 x 1 + + + . 22-1. ( )x 1 lim f x 1 = ( )x 1 lim g x ( ) ( )( ) ( )( ) ( ) 2004 f x x ln f x g x f x 4 + + = . ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( ) 2004 2004 2004 x 1 x 1 x 1 x 1 x 1 x 1 lim f x lim x lnx lim f xf x x ln f x 1 1 ln1 lim g x lim 0 f x lim f x 44 4 + ++ + + = = = = 22-2. ( ) ( ) ( )2 2 2 x 3 lim x 1 f x x g x 3 + + ( ) ( )x 3 x 3 limf x 2 limg x 1 = = . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 22 2 2 2 2 2 2 x 3 x 3 x 3 x 3 x 3 lim x+1 f x -x g x +3 lim x+1 lim f x lim limg x 3 3+1 2 -3 1 +3=-2 = + =
  9. 9. mathhmagic.blogspot.gr : . 61 22-3. : ( ) ( ) ( ) 2 2 x 2 x 1 x 0 x 0 x 0 x +1-2 . lim x +3x-2 .lim ln 2-e .lim 2 x+4 .lim x-1 . 22-4. : ( ) ( ) ( )3 7 2 x 0 x 1 x 0 3x-10 .lim x-2 2x-1 .lim x +5x-7 .lim x+5 i ii iii . 22-5. 2 x 2 x lim 2x +xx 2 . 22-6. , ( )2 2 x 2 lim x -3x+ +5 1 = . 22-7. : i) 5 3 x 0 lim(x -4x -2x+5) ii) 10 3 x 1 lim(x -2x +x-1) iii) 8 20 x -1 lim(x +2x+3) iv) 3 2 x 3 lim (x-5) |x -2x-3|[ ] v) 4 x 1 x +2x-5 lim x+3 vi) 2 2x 0 |x -3x| + |x-2| lim x +x+1 vii) 23 x 1 lim (x+2) viii) 2 2x 1 x +x+2-2 lim x +4x+3 . 22-8. f x 2 limf(x)=4 . x 2 limg(x) : i) 2 g(x)=3(f(x)) -5 ii) 2 |2f(x)-11| g(x)= (f(x)) +1 iii) g(x)=(f(x)+2)(f(x)-3) . 2. x ox 22-9. : i) 2 x 0 lim(x -3x+4) ii) x -2 lim[(2x-1)(x+4)] iii) x 4 1 lim x+ x iv) x 0 lim(2x+3x) v) x 4 lim(3x+x) . 22-10. : i) 4 3 x 1 lim(x -2x +3x+1) ii) 3 2 x 2 lim(x -2x +2x-3) iii) ( ) ( )5 x -2 lim x+3 x+4 iv) 2x 3 6-x lim x -3x+3 v) 2 x 1 lim x -2x+6 vi) ( ) ( )2 x 2 lim x -2x+1 x vii) 2 x -2 x -4 lim 3(x-2) viii) 2 2x -1 5x lim x +1 ix) x 0 lim[(x+1)x] 22-11. ( )x 1 limf x 1 = , : i. ( ) ( )( )2 x 1 lim f x -2f x +3 ii. ( ) ( )x 1 f x +3 lim f x +1 iii. ( ) ( )2 x 1 lim f x +f x +2 iv. ( )( ) ( ) ( ) 4 2x 1 f x -2 +2 lim f x +f x +1 . . f, , o x ( > 0) f + + 0 0 0 0 0 0 0 0 ( , ) ( , ) (*) ( , ) ( , ) x a x x x a x a x x x a . f (*) , x , o x o x . .
  10. 10. mathhmagic.blogspot.gr : .62 23 0 0 : ( Horner) ( ) ( )0x x x ( 0x x 0x x ). . ( 0 0 . 0x . ). 1: 3 3 2x 1 x 7x 6 lim x 3x x 3 + + . 23-1. ( ) ( ) 3 2 3 2 2x 2 x 1 x 5x 2 x x x 3 i lim ii lim x 3x 2 x 3x 2 + + + + + i). ( ) ( )2 3 x 2 x 2 lim x 3x 2 0 lim x 5x 2 0 + = + = . ( )( ) ( )( ) 23 2 2x 2 x 2 x 2 x 2 x 2x 1x 5x 2 x 2x 1 lim lim lim 7 x 3x 2 x 2 x 1 x 1 + + + = = = + . ii). ( ) ( )2 2 3 x 1 x 1 lim x 3x 2 0 lim x x x 3 0 + = + + = . ( )( ) ( )( ) 22 3 2 2x 1 x 1 x 1 x 1 x 2x 3x x x 3 x 2x 3 lim lim lim 6 x 3x 2 x 1 x 2 x 2 + ++ + + + = = = + + + . 23-2. + + 2x 2 1 4 lim x 2 x 4 , . i). x 2= , . ( )( )2x 2 x 2 x 2 1 4 x 2 1 1 1 lim lim lim x 2 x 4 x 2 x 2 x 2 2 2 4 + + = = = = + + . 23-3. 2 2x 2 x 1 2 lim x 3x 2 x 2x + . ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )x 2 x 2 x 2 x 2 x x 1 2 x 1 x 1 x 2x 1 2 1 1 lim lim lim lim x 1 x 2 x x 2 x x 1 x 2 x x 1 x 2 x 2 = = = = . : ( ) ( )( )0x x lim f x g x + ( ) ( ) 0 0x x x x lim f x lim g x + , . .. 2 2x 2 x 2 x 1 2 lim ,lim x 3x 2 x 2x + R. 23-4. 2 x 4 x -16 lim x-4 2 x -5 x -25 lim x+5 2 x 2 x -4 lim x-2 3 2x 1 x +1 lim x -1 4 3x 2 x -16 lim x -8 3 2x 2 x -8 lim x -5x+6 3 2 2x 1 x -2x +3x-6 lim x -5x+6 2 x 3 x -2x+3 lim x-3 2 x 3 x +6x+9 lim x+3 2 x 2 2x -3x-2 lim x-2 2 2x 1 2x -3x+1 lim x -1 3 2 3x 2 x -x -x-2 lim x -8 1 2 5 3 + 3 2x 2x x lim 2x x 32 8 7 6 + 3 x x lim x x 4 2 2x 2 x +x -5x-10 lim x -3x+2 3 2 3 2x 2 x -2x -3x+6 lim x +x +x-14 4 2 3 2x 1 x -2x +3x-2 lim x -4x +2x+1 23-5. 3 x 0 (x+3) -27 lim x ( )3 2x 1 x+3 -27 lim x +x x 1 2 1 1- xlim 1 1- x 2 2x 2 x +x+1 5x-2 lim + x-2 x -3x+2
  11. 11. mathhmagic.blogspot.gr : . 63 24 0 0 : 1. 0 0 , . . . 1: : x 5 x 1 x 5 2 3x 1 . lim . lim x 1 2 x 3 2 + + . 24-1. ( ) 2 2x 2 x 5 3 i lim x 2x + ( ) x 1 2x 7 3 ii lim x 3 2 + + i). 2 2 x 5 0 x 2x 0+ , ( )x x 2 0 x 0 x 2 . ( ) ( ) ( ),0 0,2 2, = + . 0 0 . x 2 ( ) 2 2 x 5 3 f x x 2x + = : ( ) ( )( ) ( )( ) ( ) 2 2 2 2 2 2 x 5 9 x 4 x 2 f x x 2x x 5 3 x x 2 x 5 3 x x 5 3 + + = = = + + + + + + , ( ) ( )x 2 x 2 2 x 2 4 1 limf x lim 2 6 3x x 5 3 + = = = + + . ii). . 7 2x 7 0 x ,x 3 0 x 3 x 3 2 0 x 1 2 + + + . [ ) ( )3,1 1, = + . x 1 ( ) 2x 7 3 f x x 3 2 + + ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )2x 7 9 x 3 2 2 x 1 x 3 2 2 x 3 2 f x 2x 7 32x 7 3 x 3 4 x 1 2x 7 3 + + + + + + + = = + ++ + + + + ( ) ( ) ( ) x 1 x 1 2 x 3 2 2 2 2 4 limf x lim 3 3 32x 7 3 + + + = = = ++ + . 24-2. ( ) ( ) 2 2x 1 x 1 x 3 3x 1 2 2x 1 x 3 i lim ii lim x 1 x 1 + + + . x 1= . 0 0 . , . i).
  12. 12. mathhmagic.blogspot.gr : .64 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) 2 2 22 2 2x 1 x 1 x 1 2 2 2 x 1 x 1 x 12 2 2 2 x 3 3x 1 x 3 3x 1x 3 3x 1x 3 3x 1 lim lim lim x 1 x 1 x 1 x 3 3x 1 2 4x 3x 1 2 x 1 4x 1 2 4x 1 lim lim lim x 1 x 3 3x 1 x 1 x 1 x 3 3x 1 x 1 x 3 3x 1 2 4 1 1 1 1 + + + + + + = = = + + + + = = = + + + + + + + + + = + 2 5 41 3 3 1 1 = + + ii). ( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( ) x 1 x 1 x 1 x 1 2 2x 1 x 3 2 2x 1 x 3 x 12 2x 1 x 3 lim lim x 1 x 1 x 1 2 2x 1 x 3 7 x 1 x 1 7 x 1 7 1 1 7 lim lim 22 2x 1 x 3 2 2 1 1 1 3x 1 2 2x 1 x 3 + + + + + = = + + + + + + = = = = + + + + + + 24-3. ( ) 2 x 1 2 x 1 x x i lim x 1 . i). x 1= , . ( ) ( )( ) ( )2 x 1 2x 1 x 1 x 1 x 1 x 1 1 xx 1 x x 1x 1 x x 1 x 1 1 lim lim lim lim 0 x 1 x 1 x 1 1 1x 1 x 1x 1 > = = = = = + + + + 24-4. ( )x 2 limf x 3 = ( ) ( )x 2 f x 3 lim f x 1 2 + . 0 0 , 0. ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) f x -3 f x +1+2 f x -3 f x +1+2 = f x -3f x +1-2 f x +1+2 . ( )( ) ( )x 2 x 2 lim f x 1 2 limf x 1 2 3 1 2 4 + + = + + = + + = . 24-5. x 4 x 2 lim x-4 x 9 3- x lim 9-x x 3 x 3 lim x-3 x 5 x- 5 lim x-5 2x 7 7- x lim x -49 2x 4 x 2 lim x 16 2x 4 x-2 lim x -5x+4 24-6. h 0 1+h-1 lim h x 3 x+1 2 lim x-3 2x 1 x+3-2 lim x -1 2 2x 0 1- 1-x lim x 24-7. x 9 9-x lim 3- x x 1 1-x lim 10-x-3 2 x 1 x - x lim x-1 x 1 x-1 lim x-1 2x 2 x+2-2 lim x +5-3 2 x 1 x +3+2x lim x+1 3x x-3 lim 1- x-2 24-8. 2 21 2 4 3 2 3 6 1 + + x x x lim x x 2 2x 1 x +x+4+x-1 lim x +3x+2 2 x 2 x +5-x-1 lim x+2-x 2 2x 3 x +2x+10-x-2 lim x -4x+3 2 3x 1 x +3x+5+x-4 lim x-x
  13. 13. mathhmagic.blogspot.gr : . 65 2. , 2 2 + + ( )( )3 3 2 2 = + + . 2: 2 3x 2 x x 2 lim x 6 2 + . 24-9. 3 2 3x 2 x +x+2-x lim x+6-2 , 3 3x 1 x 1 lim 7 x 9 x + + 3. : ,,, 3: 2 3x 3 x 7 4 lim x 5 2 + + . 24-10. 2 3 3x 1 x 1 lim 1 x 2 . ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) 2 22 3 3 3 3 3 33 3 2 3 3x 1 x 1 x 123 3 3 3 33 2 3 3 33 3 3 3 3 3 x 1 x -1 1-x + 2 1-x+ 4 x 1 x 1 1 x 2 1 x 4 x 1 lim lim lim 1 x 21 x 2 1-x- 2 1-x + 2 1-x+ 4 x 1 1 x 2 1 x 4 2 4 2 2 4 lim 6 4. 1 1 + + + = = = + + + + = = = 24-11. 3 3 3 3x 3 4 x 2x 5 lim 2x 2 3x 1 + . 0:0. 3 33 3 4 x, 2x 5, 2x 2 3x 1= = = + = ( )( )( ) ( )( )( ) ( )( ) ( )( ) 2 2 2 2 3 3 2 2 2 2 2 2 3 3 2 2 + + + + + + = = + + + + + + . ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 3 3 3 3 3 3 3 3 4 x 2x 5 4 x 2x 5 3 x 3 2x 2 3x 1 2x 2 3x 1 x 3 = = + = = + = + + = 12. 24-12. 33x 5 x 1 2 lim x 3 2x 2 + . 33 x 1, x 3 2x 2= = + = ( )( )( ) ( )( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 3 3 2 2 4 2 2 2 + + + + + = = + + + + . 22 4 x 1 4 x 1 4 x 5 = = = ( ) 3 33 3 33 x 3 2x 2 x 3 2x 2 x 5 = + = + + = .
  14. 14. mathhmagic.blogspot.gr : .66 ( )( ) ( )( ) 2 23 33 3 x 5 x 5 x 3 x 3 2x 2 2x 2 lim 3 x 5 x 1 2 + + + + = + . 24-13. . 3 x 1 x-1 lim x-1 3x 2 x 2 3x 2 lim 3 x 1 + 3 3 x 1 x 2 x lim x 3 3x 1 + 4. : () . . 2 ( ). . 4: 2 23 x 2 x x 2 x x 6 lim x 2 + + . 24-14. 3 x 2 3 x 6 2 4x 1 lim x 2 + + . 3 3 3 3 x 6 2 4x 1 3 x 6 6 2 4x 1 6 3 x 6 6 2 4x 1 6 x 2 x 2 x 2 x 2 + + + + + + + + = = . . . 3 x 2 x 2 3 x 6 6 1 2 4x 1 6 4 lim lim x 2 4 x 2 3 + + + = = . 3 x 2 3 x 6 6 2 4x 1 6 1 4 13 lim x 2 x 2 4 3 12 + + + = = . 24-15. 3 2 x 2 x 2 x 4 lim x 2 + + 3 x 2 5x 2 3x 2 lim x 2 3 2x 2 4 2x 5 6 3x 2 lim x 3x 2 + + + . 25 ( ) ( )( )0x x lim f g x . ( )u g x= . ( ) 0 0 x x u lim g x = . ( )( ) ( ) 0 0x x u u lim f g x lim f u ... = = 1: ( )2 x 0 limln x 4 + . 25-1. i). ( )x 2 limf 3x 1 7 = , ( )x 5 limf x . ii). ( ) x 0 f x lim 2 x = , ( ) x 0 f 3x lim x . i). y 3x 1= . ( )x 2 lim 3x 1 5 = . ( ) ( )x 2 y 5 limf 3x 1 limf y = . ( )x 5 limf x 7 = . ii). ( ) ( ) ( ) ( )y 3x x 0 x 0 y 0 x 0 f yf 3x f 3x f x lim lim 3 3lim 3lim 3 2 6 x 3x y x = = = = = = . 25-2. ( ) 2x 1 x x 1 i lim x x , ( ) 3 x 2 x 1 x 1 2 ii lim x 2 + . i). x t= , 2 2 4 x t ,x t= = ( ) 2 3 42 3x 1 t 1 t 1 t 1 x x 1 t t 1 t 1 1 lim lim lim lim 1 t t tx x t t 1 = = = = .
  15. 15. mathhmagic.blogspot.gr : . 67 ii). 6. 6 x 1 t = , 3 2 6 63 x 1 t , x 1 t x 1 t x t 1 = = = = + . ( )( ) ( )( ) 23 2 23 6 5 4 3 25 4x 2 t 1 t 1 t 1 t 1 t 2t 2x 1 x 1 2 t t 2 t 2t 2 5 lim lim lim lim x 2 t 1 t t t t 1 6t 1 t t ... 1 + + + + + + = = = + + + + + + + . 25-3. ( ) 0x x lim f x = . : i). f , ( ) 0x x lim f x = . ii). f , ( ) 0x x lim f x = . i). f ( ) ( )f x f x = 0x . ( ) ( ) ( ) ( ) 0 0 0 0 t x x x x x t x x x lim f x lim f x limf t lim f x = = = = = . ii). f ( ) ( )f x f x = 0x . ( ) ( )( ) ( ) ( ) 0 0 0 0 t x t x x x x x x x t x lim f x lim f x lim f x limf x = = = = = = = . 25-4. x R ( ) ( )f x 2 f x = ( )x 1 lim f x 3x 2 5 = , ( )x 3 limf x . 25-5. f: ( ) ( ) ( )f x+y =f x +f y x,y . . f . ( ) ( )x limf x =f , , ( ) 0x x lim f x , 0x . 25-6. f R ( ) ( ) x 2 f x f 2 lim l x 2 = , ( ) ( ) x 2 f x f 2 lim x 2 + . : ( )=u f x ( )=y g u ( ) = 0 0x x lim f x u , ( ) 0 f x u 0 x x ( ) = 0u u lim g u , ( )g f(x) 0 x ( ) ( ) = = = 0 0 0x x x x x x lim(gof) x lim g(f x ) lim g(u) (1) 1. ( ) 0 f x u 0 x x ; ! ( ) =f x 0 ( ) x 1,1 ( ) = = u ,u 0 g u u 2, u 0 ( = x 0 x lim 1 x ) ( ) x 1,1 ( ) = =g(f x ) g(0) 2 , ( ) = = = x 0 u 0 limg(f x ) g(0) 2 limg(u) 1 2. (1) ( ) ( ) = = 0 0 0x x x x lim g(f x ) g(lim f x ) g(u ) ; ( ) ! 0 u g = 0 g(u ) + = + = + = x 0 x 0 lim( x) (lim( x)) ( 0) 0 + = + = + = = x 0 x 0 limln(1 x) ln(lim(1 x)) ln(1 0) ln1 0
  16. 16. mathhmagic.blogspot.gr : .68 26 0 0 1. : 2, . 2, 0 0 . : 3 . 1: 2x 2 x 1 7x 2 3 lim x 4 + + . 26-1. 2x 1 x x 1 1 lim x 1 + . ( ) ( )x 1 x 1 0 limf x lim f x 0+ = = . . 2 2 2 x x 1 1 x 1 x 1 x 1 x 1 x 1 + = + . ( ) ( ) ( ) ( ) 2 2 2 x 1x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1x 1 x 1 x 1 = = = + + + + + ( )2x 1 x 1 x 1 x 1 lim lim 0 x 1 x 1x 1 + + = = + + . 2 2x 1 x 1 x 1 x 1 1 x 1 1 1 lim lim x 1 x 1 x 1 x 1 2x 1 x 1 + + = = = = + + + . 2 2 2x 1 x 1 x 1 x x 1 1 x 1 x 1 1 1 lim lim lim 0 2 2x 1 x 1 x 1 + = + = + = . 26-2. x 2 x 2 x 7 1 lim x 2 + + + . ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) x 2 x 2 x 2 x 2 x 2 x 2 2 x 2 2 x 7 3 x 7 3x+2- x+7+1 x 2 2 x 7 3 lim lim lim x-2 x 2 x 2 x 2 x 2 2 x 2 x 7 3 x 2 4 x 7 9 1 1 1 1 1 lim lim . 12x 2 x 2 2 x 2 2 x 7 3 4 2 9 3x 2 x 7 3 + + + + + + + + = = = + + + + + + = = = + + + + + + + + + + 26-3. 3 x 8 x 1 x 1 lim x 8 + . [ ) ( )x 0,8 8, + = . ( ) ( )x 7,8 8,9 ( ) ( )3 x 8 x 8 lim x 1 x 1 lim x 8 0 + = = . ( ) 3 f x 1 x 1 f x x 8 + = = ( ) ( ) ( )( ) ( )( ) 3 3 3 2 33 2 23 x+1 -9x+1-3- x+2 x+1-3 x-2 x-8 1 1 f x = - - - x-8 x-8 x-8 x+1+3x-8 x+1+3 x +2 x+4x-8 x + x 2+2 = = = .
  17. 17. mathhmagic.blogspot.gr : . 69 ( ) 3 3 32 6 3x 8 x 8 3 1 1 1 1 1 1 1 limf x lim - 6 12 12x+1+3 9 3x +2 x+4 2 +2 2 +4 = = = = + . 26-4. x 2 3x 2 x 1 3 lim x 2 + . 26-5. x 1 3x 1 x 3 lim x 3 2 + + + . 2. : , .. 2,3, ( 0 0 ) . 2: x 4 3x 3 2 x 5 2x 1 lim x 2 + + + . 26-6. 2x 3 2x 3 2 3x x 6 lim x 5x 6 + + + + . ( )( ) ( )( )2 2 2x 3 2 3x x 6 2x 3 3x x 6 3x 2x 3 3x x 6 3x x 5x 6 x 5x 6 x 3 x 2 x 3 x 2 + + + + + + + + = = + + + . ( )( ) ( )( )x 3 x 3 2x 3 3x 1 x 6 3x 1 lim lim x 3 x 2 6 x 3 x 2 3 + + = = . ( )( ) ( )( )x 3 2x 3 3x x 6 3x 1 1 1 lim x 3 x 2 x 3 x 2 6 3 2 + + + = = . 26-7. x 1 2 x 3 6 2x 3x 1 lim x 1 + + . 26-8. 3 3 3 2x 1 2 x 2 x 3 2x lim x 3x 4 + + . ; o (Qr-code)
  18. 18. mathhmagic.blogspot.gr : .70 27 0 0 H 1. 2 A A A= = . = . 1: x 2 x 2 lim x 2 x 1 + 27-1. x 3 x 3 lim x 3 12x + + . x 3+ x 3, x 3 0> > ( ) 2 x 3 x 3 x 3 = = . ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) + + + + = = = = + + + + + + 2 22 2 2 x 3 x 3 12x x 3 x 3 12xx 3 x 3 x 3 12x x 3 12x x 3 12x x 3 12xx 3 12x ( ) ( ) ( ) + + = = + + 2 2 x 3 x 3 12x x 3 12x. x 3 x 3 lim x 3 12x 3 3 12 3 2 3+ + + = + + = . 27-2. x 7 x 7 lim x 1 3 x 3 + . 27-3. 2 x 0 1 1 x lim x+ . 2. . 0 0 . 2: x 1 x 1 lim x x 8 4 + + . 27-4. x 1 x 1 lim 3 x x 3 5 + + . . 0 0 . . ( )3 x 13 x x 3 5 3 x 3 x 3 2 x 3 2 x 1 x 1 x 1 x 1 + + + + + = = + . : ( ) x 1 x 1 3 x 1 3 x 3 2 1 lim lim x 1 2 x 1 4 + = = . x 1 3 x x 3 5 3 1 7 lim x 1 2 4 4 + + = + = , x 1 x 1 4 lim 73 x x 3 5 = + + . 27-5. x 1 x 1 lim x x 3 3 + + . 27-6. 2 x 1 x 1 lim 5x 1 2x 1 x 3 5 + + +
  19. 19. mathhmagic.blogspot.gr : . 71 28 0 0 . ... . . ( )y g x= y. ( )y g x= 0x x . y. 1: 3 3x 0 3 2x 1 2 2x 1 1 lim 2 2x 1 2x 1 1 + + + + . 28-1. 3 5x 1 1 x lim 1 x . x 0 x 1 , ( ) 3 5 1 x f x 1 x = 1. 15 x t= , 5 33 5 x t x t= = . x 1 t 1, t 1 . ( )( ) ( )( ) 2 3 45 2 3 43 3 225 1 t 1 t t t t1 x 1 t 1 t t t t 1 t 1 t t1 x 1 t 1 t t + + + + + + + + = = = + + + + . 2 3 43 25x 1 t 1 1 x 1 t t t t 5 lim lim 1 t t 31 x + + + + = + + . 28-2. 6 3 6x 0 3x 1 3x 1 lim 3x 1 3x 1 + + + + . . . . 2,3,6 6. 6 3x 1 y+ = , 3 23 3x 1 y 3x 1 y+ = + = . x 0 6 6 x 0 lim 3x 1 0 1 1 + = + = , y 1 . ( )( ) ( ) ( ) 3 2y 1 y 1 y 1 y y 1 y 1y y lim lim lim y 1 2 y y y y 1 + = = + = . 28-3. ( ) ( ) 5 x 1 x 0 x x 1 x 1 1 i lim ii lim xx x 2 + + i). x t 1= , 2 x t= ( )( ) ( )( ) 23 2 2x 1 x 1 x 1 x 1 t 1 t t 1x x 1 t 1 t t 1 3 lim lim lim lim 1 t t 2 t 1 t 2 t 2 3x x 2 + + + + = = = = = + + ++ . x 1 x x 1 lim 1 x x 2 = + ii). 5 x 1 t 1+ = , 5 x t 1= ( )( ) 5 5 4 3 2x 0 x 0 x 0 x 1 1 t 1 t 1 1 lim lim lim x t 1 5t 1 t t t t 1 + = = = + + + + . 5 x 0 x 1 1 1 lim x 5 + = . 28-4. : 3 4 4x 0 x+ x+ x lim x+ x+ x 3 4 34x 1 2 x 3 x x lim 3 x 2 x x + 32 2 3 2x 0 x 1 x 1 lim x 1 1 + + + 3 6 3 6x 0 1 x 1 x 5 1 x 5 lim 1 x 1 x 2 + + + + + + + 3 6 3x 9 7x 1 3 7x 1 4 7x 1 4 lim 7x 1 4 + + + + +
  20. 20. mathhmagic.blogspot.gr : .72 29 : . : 0 0x x , x x + . 0x . 0x . : x , x< > , .. x . 1: ( ) ( )x 1 x 2 limf x limf x : ( ) 2 3 x 3, x 1 f x 2x 2, 1 x 2 x 2x 4, 2 x + < = < < < . 29-1. ( ) 2 x 2, x 1 f x 3x 4, 1 x 2 2x 1, x 2 = < < > . ( ) ( ) ( ) ( ) ( ) ( ) ( )x 0 x 1 x 2 i limf x ii limf x iii limf x i). 0 ( ) 2 f x x 2= ( ) ( )2 2 x 0 x 0 limf x lim x 2 0 2 2 = = = . ii). f 1 . x 1< , ( ) 2 f x x 2= , ( ) ( )2 2 x 1 x 1 lim f x lim x 2 1 2 1 = = = . 1 x 2< < , ( )f x 3x 4= , ( ) ( ) x 1 x 1 lim f x lim 3x 4 3 1 4 1+ + = = = . ( ) ( ) x 1 x 1 lim f x lim f x 1 + = = ( )x 1 limf x 1 = . iii). ( ) ( ) x 2 x 2 lim f x lim 3x 4 3 2 4 2 = = = . ( ) ( ) x 2 x 2 lim f x lim 2x 1 2 2 1 3+ + = = = . ( ) ( ) x 2 x 2 lim f x lim f x + ( )x 2 limf x . 29-2. : ( ) ( )x 0 x 2 .limf x .limf x ( ) 2 3 x +2, x 2 f x = x -3x+4 , x>2 x-1 . 29-3. ( ), f 0x : i) 2 x , x 1 f(x)= 5x, x>1 0x =1 ii) 2 -2x, x 0x ( fD ). ( ) 0x x lim f x 0 < ( )f x 0< 0x . f, g 0x ( ) ( )f x g x 0x ( ) ( ) 0 0x x x x lim f x lim g x . ( )f x 0> 0x ( ) 0x x lim f x 0 . ( )f x 0< 0x ( ) 0x x lim g x 0 . ( ) ( ) 0 0x x x x lim f x lim g x > ( ) ( )f x g x> 0x . ( ) ( )f x g x< 0x ( ) ( ) 0 0x x x x lim f x lim g x . 1. : 0 0 . 1: x 1 x 1 7 lim x 1 3x + + + . 30-1. 3 2x 2 x x 1 x 7 lim x 4 ( )3 x 2 lim x x 1 5 0 = > 3 x x 1 0 > 2. 3 3 x x 1 x x 1 = 2. ( )x 2 lim x 7 5 0 = < x 7 0 < 2, x 7 x 7 = + . ( )( ) ( )( ) 3 23 3 2 2 2 2x 2 x 2 x 2 x 2 x 2 x -x-1 - x-7 x-2 x +2x+4x -x-1+x-7 x -8 x +2x+4 lim lim lim lim lim =3 x -4 x -4 x -4 x-2 x+2 x+2 = = = = . 30-2. 2 x 2 3 x-2 + x -5 -1 lim x -3 2. 0 0 : . , 0x ( ). 0x . , 0 0 . 2: 2 x 2 x-3 +2 x -1 -7 lim x-2 . 30-3. + 2 2x 2 x 4 x 2 lim x 2x
  21. 25. mathhmagic.blogspot.gr : . 77 ( ) 2 2 x 4 x 2 f x ,x 0,2 x 2x + = . 2 x 4 0 x 2 = = . x -2 2 ( )f x + - + ( )x 0,2 ( ) 2 2 2 2 x 4 x 2 x x 6 x 3 f x x 2x x 2x x + + + = = = . x 2> ( ) 2 2 2 2 x 4 x 2 x x 2 x 1 f x x 2x x 2x x + + = = = . ( ) ( ) x 2 x 2 x 2 x 2 x 3 5 x 1 3 lim f x lim lim f x lim x 2 x 2 + + + = = = = ( ) ( ) x 2 x 2 lim f x lim f x + ( )x 2 limf x . 30-3. 3 x 1 x -x+2 -2 lim x-1 2 2x 1 x +x+2 + x+3 -8 lim x +x -x-1 2 x 1 x+2 + x -3 -5 lim x -1 2 2x 2 x + x-3 -2x-1 lim x -4 x 3 2 x 1 3 x 5 2 lim x 1 5 x + 3. 0 0 : 0 0x x , x x+ . 0x . ( ). ( , 0x ). 3: 2 2x 2 x-2 +x -4 lim x -4 . 30-4. , , 2x 3 2 x-3 +5 1-x -10 lim x -9 2 x 2 x x 4 x 2 2 lim x 2 2 x 5 |x-5| +x -4x-5 lim x-5+ .. f,g ( A ). 0 x A : , ( ) ( ) 3x x f x 3x f x x , ( ) x 0 x 0 x 0 3x 3x x 3x lim f x lim lim 3 x 3x x x+ + + (1). ( )x ,0 . x 0< , ( ) ( ) 3x x f x 3x f x x , ( ) x 0 x 0 x 0 3x 3x x 3x lim f x lim lim 3 x 3x x x (2). (1) (2) ( )x 0 limf x 3 = = . 31-9. : x 0 3x lim x x 0 x lim x 3x 0 x lim x +x x 0 x-x lim x x 0 2x lim 3x x 0 4x lim 2x x 0 x lim 2x x 0 5x lim 5x+4-2 x 0 3x-x lim 6x-2x 31-10. : 2 x x lim 1+x 2 x 0 1- x lim 2x 2x 0 1-2x lim x 2x 0 x lim x -x x 0 x+x lim 1+x 1 x 0 x-3x lim x-3x 31-11. x 0 2 3+x-2 lim 1+ x-1 2 2 2 2x 0 3x - x lim x + x 31-12. =1. =1 : i) 2 lim(-) , ii) 2 2 2 lim( - ) iii) 2 lim .
  22. 29. mathhmagic.blogspot.gr : . 81 31-13. f ( ) 2 f x x-3x x , x . ( ) ( )x 0 x 0 6x+2x lim f x lim xf x +x = =A B . 31-14. f : i. ( )x 0 lim f x L = ii. ( ) 2 xf x x, x . L. 31-15. x 0 x+2x 3x , +2=3 . 31-16. x 0x . ( ) 0x x lim f x = , ( )f x 0< 0x . ( ) 0x x lim f x = + , ( )( )0x x lim f x = . ( ) 0x x lim f x = + , ( )0x x 1 lim 0 f x = . ( ) 0x x lim f x 0 = ( )f x 0> ( )0x x 1 lim f x = + . ( ) 0x x lim f x 0 = ( )f x 0< ( )0x x 1 lim f x = . ( ) 0x x lim f x = + , ( ) 0x x lim f x = + . 0 2x x l lim * x = . : ( ) 00 , , , 0 , , 0 , 1 0 + + ..
  23. 32. mathhmagic.blogspot.gr : .84 33. ( )0x x . f 1 f 0 g g = 0 + 0 0 0 + ( ) ( )0x x f x lim g x 0 1). ( ) 0x x lim f x = . 2). ( ) 0x x lim g x 0 = , ( ) ( ) ( ) ( ) f x 1 f x g x g x = . 3). ( )g x 0x ( ). ( )g x 0> ( )0x x 1 lim g x = + . ( ) ( ) ( ) ( ) ( ) 0 0 0x x x x x x , 0f x 1 lim lim f x lim , 0g x g x + > = = + < . ( )g x 0< ( )0x x 1 lim g x = . ( ) ( ) ( ) ( ) ( ) 0 0 0x x x x x x , 0f x 1 lim lim f x lim , 0g x g x > = = + < . 1: ( ) + 2x 1 3x 5 lim x 1 2x 3 x+2 lim x -9 2 2x 3 x x 6 lim x 3 x 5x 6 + : 0 0 , 0 . 33-1. : ( ) ( ) ( ) ( ) 2 2 x 2 x 2 x 2 x 2 5 x 5x 5 x 5x lim lim lim lim x 2 x 2 x 2 x 2 + + { }R 2 . (). x 2< ( ) x 2 lim x 2 0 x 2 x 2 0 = < < . x 2 5 lim x 2 = . (). x 2< ( ) x 2 lim x 2 0 x 2 x 2 0 = < < . ( )2 x 2 lim x 5x 4 10 6 0 = = < . 2 x 2 x 5x lim x 2 = + . (). x 2> ( ) x 2 lim x 2 0 x 2 x 2 0 = > > . x 2 5 lim x 2+ = + . (). x 2> ( ) x 2 lim x 2 0 x 2 x 2 0+ = > > . ( )2 x 2 lim x 5x 4 10 6 0 = = < . 2 x 2 x 5x lim x 2+ = . 33-2. ( ) ( )x 0 x 0 2x 3 3x 2 lim lim xx x x + .
  24. 33. mathhmagic.blogspot.gr : . 85 (). ( )x 0 x 0 2x 3 1 lim lim 2x 3 xx xx = . x 0 lim xx 0 0 0 x,x = = 0, xx 0> , x 0 1 lim xx = + . ( )x 0 lim 2x 3 2 0 3 3 0 = = < , ( ) ( )x 0 2x 3 lim 3 xx = + = . (). ( )x 0 lim x x 0 0 0 x x x x 0 = = < < 0, x 0 1 lim x x = . ( ) ( ) ( )x 0 x 0 3x 2 1 lim lim 3x 2 3 0 2 x x x x + = + = + = . 33-3. ( ) 2x 1 3x 2 lim x 1 + . ( ) 2 3x 2 f x , x 1 x 1 + = . ( ) ( )( )2 3x 2 3x 2 3x 2 1 f x x 1 x 1 x 1 x 1 x 1 + + + = = = + + . : ( ) ( ) x 1 0 x 1 x 1 3x 2 1 5 lim f x lim x 1 x 1 2+ + > + = = + = + + . ( ) ( ) x 1 0 x 1 x 1 3x 2 1 5 lim f x lim x 1 x 1 2 < + = = = + . ( ) ( ) x 1 x 1 lim f x lim f x+ f 1. 33-4. , , 2 2x 1 x 5x 6 lim x x + . ( ) ( )2 2 x 1 x 1 lim x x 0 lim x 5x 6 2 = + = x 0 ( ) ( ) 2 2 x 1 x 1 x 1 x 5x 6 x 5x 6 1 limf x lim lim x x 1 x x 1 + + = = . 1 ( )x 1 , , ( ) 2 x 1 x 1 x 5x 6 1 lim f x lim x x 1+ + + = = + 2 x 1 x 1 x 5x 6 1 lim 2 lim x x 1+ + + = = + . ( ) 2 x 1 x 1 x 5x 6 1 lim f x lim x x 1 + = = 2 x 1 x 1 x 5x 6 1 lim 2 lim x x 1 + = = . ( ) ( ) x 1 x 1 lim f x lim f x+ . f 1. 33-5. 3x 0 2 x 2 x lim x + . ( )3 x 0 x 0 lim x 0 lim 2 x 2 x 0 = + = . 0 f ( ) ( )( ) ( ) ( ) ( ) 23 3 3 2 x 2 x 2 x 2 x 2 x 2 x 2x 2 1 f x x2 x 2 xx 2 x 2 x x 2 x 2 x x 2 x 2 x + + + = = = = + + + + + + + + ( ) 2x 0 x 0 2 1 limf x lim x2 x 2 x = = + + 2x 0 x 0 2 2 1 lim lim x2 x 2 x 2 2 = = + + + .
  25. 34. mathhmagic.blogspot.gr : .86 33-6. 2x 1 1 lim x 1 . ( )2 x 1 lim x 1 0 = . 2 x 1 x 1 : x -1 1 + 2 x 1 + - + 2 2x 1 x 1 1 1 lim lim x 1 x 1 + = = + . 2x 1 1 lim x 1 . 33-7. 2 2x 2 2 x lim x 4 x 2x . ( )( ) ( ) ( )( ) ( ) ( ) +2 2x 2 2 2x 2 x 2 x 2 2 x 2 1 -x 1 lim = + - =- 2 x 2 x -x x-2 x +2x 2 lim - lim - =lim 1 -x 1x -4 x -2x x-2 x+2 x x-2 x x-2 x+2 lim = - =+ x-2 x +2x 2 = . 33-8. , , : ( ) ( ) ( )2 2x 0 x 0 x 0 1 1 1 1 1 1 lim lim lim x x x x x x + . (). 2x 0 x 0 1 1 lim lim x x = + = + . 2x 0 1 1 lim x x + = + . (). x 0 x 0 x 0 1 1 1 1 lim lim lim 0 0 x x x x+ + + = = = . x 0 x 0 x 0 1 1 1 1 2 lim lim lim x x x x x = = = + . ( ) 1 1 f x x x = 0. (). 22x 0 x 0 x 0 1 1 1 1 1 1 lim lim lim 1 x x x x xx = = = + x 0 x 0 1 1 lim lim 1 x x = + = + . 33-9. ( ): 4x 0 1 lim x 3000x 0 -1 lim x ( ) 2x 2 -7 lim x+2 ( ) 2x 1 x lim x-1 2x 2 x-3 lim x -4x+4 x 2 x -3 lim x-2 2x 3 x+4 lim x +6x+9 33-10. ( ): + x 1 2x 6 lim x 1 ( ) 2003x 1 5 lim x-1 2 2x 2 x -1 lim x -5x+6 x 4 1 lim x+ x 6 2 x 3 x -4x+1 -3 x-2 lim x-1 -2 33-11. ( ) 2 3 2 x -1 f x = x -3x +3x-1 . ( )x 1 limf x .
  26. 35. mathhmagic.blogspot.gr : . 87 33-12. ( ): 3 x 0 2x +5 lim 1-x x 2 2x-3x lim 1+x x 1 1 lim lnx 2 x 4 2x+1 lim 2 x-1 2x 0 1 lim x x 2 1 lim x 33-13. ( ): 3 x x x x 2 2 lim x lim x lim x lim xA B + + = = = = . 33-14. N ( ) f 0x : 1 1 f(x)= - x |x| 0x =0 , 2 3 4 f(x)= - 1-x 1-x 0x =1, ( ) 2 4 1 f x = - 1-x 1-x 0x =1 4 2 x+5 f(x)= x +3x 0x =0 , 4 2x-3 f(x)= 4(x-1) 0x =1 34 , .. ( )x lim f x + ( )fD ,= + . x x x x x 1 lim x lim 0, * x , 1 lim x lim 0, * , x , 0 lim x x , 0 + + + = + = + = = + > = < 1: . 1: ( )3 2 3x x x 1 lim 3x , lim 2x , lim x+ + ( )3 x lim 3x 2x 6 + 34-1. ). ( )3 x lim 2x 3x 1 + + , ) ( )2 x lim 3x 5x 1 + , ) ( )( )3 x lim x x 2 , 0 + + + < ). ( ) ( ) ( )3 3 x x lim 2x 3x 1 lim 2x 2 + + + = = + = . ). ( ) ( ) ( )2 2 x x lim 3x 5x 1 lim 3x 3 + = = + = . ). , 0< . 0< 0< + < . ( )( ) ( )( )3 3 x x lim x x 2 lim x + + + = + = . 34-2. ( )3 x lim 2x x x 1 + . ( )3 3 x x lim x x 1 lim x + + = = + , x > 3 x x 1 0 > . ( ) ( ) ( ) x 3 3 3 3 x x x x lim 2x x x 1 lim 2x x x 1 lim x 3x 1 lim x > + + + + = = + + = = . 34-3. : ). ( )5 2 x lim 4x x 3x 1 + + + , ) ( )3 2 x lim 2x 3x 1 + , ). ( ) ( )5 2 5 x x lim 4x x 3x 1 lim 4x + + + + = = . ). ( )3 2 3 x x lim 2x 3x 1 lim 2x + = = .
  27. 36. mathhmagic.blogspot.gr : .88 34-4. : ( ) 5 2 x + lim 2x +3x +x+1 ( ) 3 2 x lim 5x +2x +x-3 3 x lim (-10x +2x-5) + 3 x lim (5x -2x+1) 2: . 2: ( ) 3 2 2x 3x 2x 6x 6 lim x 1 4 + + + . : . ( ) . 3 3 3 2 2x x 1 x 2x 1 lim x 1 x 1 + + + + . : ( ) ( ) ( ) f x P x g x = . f g . f g R* ( ). f g 0. . 34-5. ). 3 3 2x 2x x 1 lim 3x x 1+ + + + , ) 2 3x x x 2 lim 2x x 1 + + + , ) 2 2 x x 2x lim x 1 x 1+ + ). 3 3 3 2 3x x 2x x 1 2x 2 lim lim 3x x 1 3x 3+ + + = = + + . ). 2 2 3 3x x x x x 2 x 1 1 1 lim lim lim 0 0 2x x 1 2x 2 x 2 + = = = = + + . ). ( ) ( ) ( ) 2 2 3 2 3 2 2x x x x x 2x x 3x x 1 x lim lim lim lim x x 1 x 1 x 1 x + + + + + + + = = = = + . 34-6. : ) 2 5 3x 2x x 1 lim 4x x 2+ + + , ) 4 2 2x 3x x 2 lim 2x x 1 + + , ) 2 3 x x 1 x 2 lim x x 1 + + + . ). 2 2 5 3 5 3x x x 2x x 1 2x 2 1 lim lim lim 0 4x x 2 4x 4 x+ + + + = = = + . ). ( ) 4 2 4 2 2 2x x x 3x x 2 3x 3 lim lim lim x 2x x 1 2x 2 + = = = + . ). ( ) 2 3 4 3 2 4 2 2 2x x x x x 1 x 2 x x x x 1 x lim lim lim lim x x x 1 x x x + + + + + = = = = + + . 34-7. ). 2 2x x 3x 2x lim x 5x 6 + + , ) x x 2 3 x 1 lim x 7+ + . ). 2 x 3x . x x 0< , 2 x 3x 0 > x 0 3 + 2 x 3x + - +