مجلة الرائد في الدوال اللوغاريتمية مرفقة بالحلول -نسخة 2019-2020

f '(x)  g(x) f '(x) x²  1 .x  1.ln x  x ²  2 1  ln x  g(x)  x  f '(x)  1  2   x² x² x² f f '(x) X0   f '(x)  g(x) f '(x) f'(x) 0 x² f (x)   g(x) f f () () (C ) (C ) yx2 () 2 f f limf (x)  (x  2)  0 (C ) y  x  2 () f x lim ln(x)  0 lim f (x)  (x  2)  lim 2ln(x)  0 xx x x x f (x)  (x  2)  2 ln x () (C ) f x x0 1  () 0 (C ) (C ) () f f () (C ) f (T) () (T) (C ) f f '(x )  1 1 (T) () (T) (C ) 0 f x e 2  2ln(x )  0 g(x )  x2 g(x ) 1 f '(x )  1 0 0 00 0 0 x² 0 24 0202

y  x 2 2 y  f '(e)(x  e)  f (e) (T) e f xx f(x)  0 f 21 f (0,6)  f(0,7) 0 0; f (x )  0 0,7 0,6 α 1 f (2,7)f(2,8) 0 ; f(x )  0 2,7 2,8 β 2 y (C ) (T) () f 4 (C ) f () 3 2 (T) 1 0 123456 x -1 -2 (m  2)x  2ln(x)  0...(1) 3 y  x  m x  m  f (x) m  2  2ln(x) 1 y  f (x) x (C ) y  x  m f 1 m  2  2 2 1 m 2  2 1 e e 1 m  2 4 1 2  2 m 2 3 e 26 0202

0244 34 ba 4 f '(1)  0 1  2 (C ) A( ;1) f (1)  1 f2 2 b .4x²  8x(a  b ln(2x) b  2(a  bln(2x) f (x)  a  bln(2x) x 4x3 4x² f '(x)  4x²²  a 1 f (1)  a  1 f (1)  1 21 2 b  2 b  2a b  2a  0 f '(1)  0 2 lim g(x) lim g(x) 0 x 0 x g(x)  1 2ln(2x) 0; g 4x² lim ln(2x)  0 limg(x)  lim 1  lim 2ln(2x)  0 x (2x)² 4x² (2x)²x x x lim 4x²  0 lim 1 2ln(2x)   lim g(x)  lim 1 2ln(2x)   x 0 x 0 x 0 x 0 (2x)² g g '(x) 2 .4x²  8x(1 2 ln(2x )) 8x(1  1  2 ln(2x ))  ln(2x) x g '(x)    (4x²)² 16x 4 x3  ln(2x)) g '(x) 22 0202

x1 g'(x)  0 x  1 g'(x)  0 x  1 g'(x)  0 2 2 2 x0 1 g f'(x) 2  f (x) 0 g(x)  0 0;  f(1) 2 (1 2ln(2x))  0 g(x)  0  x 1 ln(2x)   1 0 2e 2 y (C ) 1 f (C ) x 1 f 2e 0 1 2 3x -1 0242 36 x1  x g g'(x) g(x) g(x)  2x  ln x 1;  g 2 limln x   limg(x)  lim2x  limln x   x x x x g  g'(x) 0 g '(x)  2  1 x 1;  g  g 22 0202

g(x)  0 x 1; g(x)  2 2; g 1; g g(x)  0 x 1; 6ln x 0 f (x)  x 2  ln x x 6ln x 6ln x f (x)  x  x  6ln x . x  6ln x 2  ln x 2x  ln x x 2x  ln x 2x  ln x xx lim f (x) x lim ln x  0 lim f (x)  0  0 f (x) xx x 20 ff 6 (2x  ln x)  (2  1 )(6ln x) 12  6ln x  12ln x  6ln x 12(1 ln x) x x x x x f '(x)    (2x  ln x)² (2x  ln x)² x(2  ln x)² x1 e  f '(x) f'(x) 0 f (x) 1 ln x f (e) 0 f '(x) 0 x  e f '(x)  0 0 x e f '(x) 0 x e 1;  f f (x)  k k x 1; 0 f (x) f (e) f 22 0202

f (e)  6 k   0; 6  0 k f (e) 2e 1  2e  1 ( ) 4 (C ) ( ) 1 f1 3 y  3x  3 y  3(x 1) y  f '(1)(x 1)  f (1) h h h(x)  f (ex ) 1; h(1)  f (e)  6 limf (x)  0 limex   limh(x)  limf (ex )  0 2e  1 x x x x h '(x)  exf '(ex )  12(1 x) (2ex  x)² x1  1; (1 x) 0 h '(x) 0 h'(x) 0 1;  h h(x) h(1) 0 y 6 y  0(x 1)  (C ) ( ) ( ) 2e 1 h2 2 y y  h '(1)(x 1)  h(1) (C ) (C ) ( ) ( ) 3 h f 21 ( ) 1 ( ) 2 (C ) 2 1 f -2 -1 0 1 2 3 4 5 6x -1 -2 (C ) h 422 0202

0219 11 lim f(x) lim f(x) lim f(x) 1 x 2 x 2 x 0 f (x)  1  ln x 0;2  2; f x2 lim f(x)  lim 1  lim ln x   1  lim ln x   x 0 x  2x 0 x 0 2 x 0 lim f(x)  lim 1  lim ln x  lim 1  2   x 2 x  2x 2 x 0 x 0 x  2 lim f(x)  lim 1  lim ln x  lim 1  2   x 2 x  2x 2 x 0 x 0 x  2 (C ) x0 lim f(x)   f x 0 lim ln x (C ) x  2 lim f(x)   x  f x 2 lim f(x) x lim 1  0 lim f(x)  lim 1  lim ln x   x  x  2 x  2x  x  x  0;2  2; f 0 f '(x)  1  1  x  (x  2)²  x²  5x  4 0;2  2; (x  2)² x x(x  2)² x(x  2)² x²  5x  4 f '(x) x²  5x  4 x  4 x 1 x²  5x  4  0 x²  5x  4  0 1;2  2;4 x²  5x  4  0 0;1  4; f 1;2  2;4 0;1  4; x0 1 2 f  0 f '(x) 4 0 1   f (x)   1  2ln 2  1 0202 2

lim f(x)  ln x 3 x  lim f(x)  lnx  lim 1  0 x  x  x  2  (C ) ln () lim f(x)  lnx  0 f x  f(x)  lnx () (C ) f () (C ) f (x  2) f(x)  lnx  1 x2 () (C ) (x  2)  0 0;2 () (C ) f f (x  2)  0 0;2 2; (C ) () 4 f y (C ) 3f 2 1 () 0 1 2 3 4 5 6 7 8 9 10 x -1 -2 -3 g 6 g g(x)  f (2x) ;1  1;0 g f x  (2x) g'(x)  0 g'(x)  0 g'(x)  2f '(2x) ;1  1;0 x g'(x)  0 x  2 x  1 2x  4 2x 1 f '(2x)  0 2 f '(2x)  0 g x  2; 1  1;  1  2  g x  ; 2   1 ;0 f '(2x)  0 2 0 0202

0218 10 g(x) x g(1) I g(x)  1  (ln x)²  ln x 1 0; g x 1 (Cg ) g(1)  0 x 0;1 g(x)  0 0;1 (Cg ) x 1; g(x)  0 1; (Cg ) lim f (x)  0 limf(x) 1 II x x 0 f (x)  1 ln x 0; f 1 x ln x lim x ln x  0 limln x   limf(x)   x 0 x 0 x 0 lim 1  0 lim ln x  0 1  ln x xx xx x x lim ln x   f (x)  1  ln x lim f (x)  0 x x x x  0 (Cf ) limf(x)   x 0 y0 (Cf ) lim f (x)  0 x f '(x)  g(x) :0; ‫ من‬x 0 (1 x ln x)² ( 1 (1 x ln x)  (ln x 1)(1 ln x)) 1  ln x  (ln x)²  2ln x 1 g(x) f '(x)  x x  (1 x ln x)² (1 x ln x)² (1 x ln x)² f f g(x) f '(x) f '(x)  g(x) f 1;  (1 x ln x)² 0 x 0;1  f '(x) 1 0 1 f (x) 0  3 0202

(C ) (T) y  e² x  e 3 f e1 e1 x  1 1 ln x  0 f (x)  0 (C ) 0e f y  f '(1)(x  1)  f (1) x (T) eee 0 f (1)  0 f '(1)  g(e1)  e 1  e² e e (1 e1 ln e1)² (1  e1)² e 1 y  e² (x  1)  e² x  e e 1 e e 1 e 1 (C ) (T) f y 2 () 1 (C ) f -1 0 -1 1 2 3 4 5 6 7x -2 -3 (e 1)f(x)  e²x  em m 4 f(x)  e² x  e m (e 1)f(x)  e²x  em e 1 e 1 (T) y  e² x  e f(x)  y f(x)  e² x  e m=1 e1 e1 e1 e1 m 1 (e 1)f(x)  e²x  em 4 0202

0217 13 f1 f (x)  2 (x)  ln  x 1 D  ;1  1; f 3  x  1  f (x)  f (x)  0 x  D f f f (x)  f (x)  2 (x)  ln  x 1  2 (x)  ln  x 1 xD 3  x  1  3  x  1  f  ln  x 1  ln  x 1  ln  (x 1)(x 1)   ln1  0  x  1   x  1   (x 1)(x 1)  Of limf (x) limf (x) limf(x) lim f(x) 2 x x x 1 x 1 lim  x  1   0 lim f(x)   2  lim ln  x 1     x  1  3  x 1  x 1 x 1 x 1 lim  x  1    limf(x)  2  lim ln  x 1    x  1  3  x  1  x 1 x 1 x 1 lim ln  x  1   ln 1  0 lim 2 x   limf (x)    x  1  3x x x lim ln  x  1   ln 1  0 lim 2 x   limf (x)    x  1  3x x x (C ) f limf(x) lim f(x) x 1 x 1 x  1 (C ) f x 1 (C ) f f '(x)  2  x²  2  : D x 3 3  x²  1  f (x)  2 (x)  ln x 1  ln(x 1) D x 3 f '(x)  2  x 1  x 1 1  2  2  2  3.2  2  x² 2  3 1  3 x² 1 3 3(x² 1) 3  x² 1  f f '(x) f '(x)  2  x²  2  3  x² 1  5 0202

x f x²  2 D x² 1 f '(x)  1 1 f  f (x)      4 1,8  1,9 f (x)  0 3 limf (x)   lim f (x)   1; f 6 x x 1 f (1,9)  0.08 0 f (1,8)  0.05 0 1,8  1,9 f ()  0  (C ) y 2x () f3 limf (x)  y  0 (C ) y 2x () f 3 x  lim f (x)  y  lim ln  x 1   ln1  0  x 1  x  x  () (C ) f f (x)  y  ln  x  1  () (C )  x  1  f x 1 1 x ;1 f (x)  y 0 x 1 x 1 1 x 1; f (x)  y 0 x 1 x  1 1  f (x)  y () (C ) () (C ) f f (C ) () f 6 0202

6y 5 4 () 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8x (C ) -1 f -2 -3 -4 -5 -6 (2  3 m )x  3ln  x 1  0 7  x  1  (2  3 m )x  3ln  x 1  0....(*)  x  1  2 x  ln  x 1  m x (2  3 m )x  3ln  x 1   0 (*) 3  x  1   x 1  (*) y  m x m x  f (x)  y  f (x) y mx ( ) (C) m m2 m 2 2 2m 2 1 33 33 1I 0217 14 f lim f (x) lim f (x) lim f (x)   x x 1 x 1 2 2 f (x)  1 2ln(2x 1)   1 ;   (2x 1)² 2  lim (2x 1)²  0 lim 1 2ln(2x 1)  1 x 1 x 1 22 7 0202

lim ln(2x 1)  0 lim f (x)  lim 1  2 lim ln(2x 1)  0 x (2x 1)² x x (2x 1)² x (2x 1)² x 1 (Cf ) lim f (x)   2 (Cf ) y0 x 1 2 lim f (x)  0 x f '(x)  8ln(2x 1)   1 ;   x 0 (2x 1)3  2 2 2 (2x 1)²  4(2x 1)(1 2ln(2x 1) f '(x)  (2x 1) (2x 1)²²  41 (1 2ln(2x 1)  8ln(2x 1) (2x 1)3 (2x 1)3 f ln(2x 1) f '(x) f '(x)  8ln(2x 1) (2x 1)3 f '(x) x  0 ln(2x 1)  0 x 1 0  2 0 f '(x) f x 1 0  2 0 f '(x) 1 f (x) 0  f (x) f (x)  0   1 ;   3  2 f (x)  0 f (x) 0 x  1 (1 1 ) 1 2ln(2x 1)  0 2e x   1 ; 1 (1  1 )  1 2ln(2x 1) 0 2 2 e  8 0202

x   1 (1  1 );   1 2ln(2x 1) 0 f (x) 0  2 e f (x) f  (Cf ) 4 x0 f'' (x0; y0) (Cf )  8 2  1)3  6(2x 1)2(8ln(2x 1)) 161 3ln(2x  1) (2x 1)6 f ''(x)  (2x  2x 1 (2x 1)4 (1 3ln(2x 1)) f ''(x) f ''(x) x  1 (1 3 e) (1 3ln(2x 1))  0 2 x   1 (1  3 e );   f ''(x) 0 x   1 ; 1 (1  3 e )  f ''(x) 0  2 2 2 (Cf ) y0  f (x0 )  5 e 2 (x0; y0 ) 3 3 (C ) f y 3 2 1 (Cf ) -1 0 123456x -1 -2 -3 g 1 II g(x)  2x  ln(2x 1)   1 ;   g 2  g '(x) g g '(x)  2 1  2   2 2x 1  2x  1 2x 1  2x 1 g '(x) 9 0202

x   1 ;   g '(x) 0 x    1 ; 1  g'(x) 0 x  1 g'(x)  0  2   2 2  2 1,2  1,3  g(x)  0 g(1,3)  0,03 g(1,2)  0,047  1 ;   g  2 g(x)  0 1,2  1,3 g()  0  g(0)  0 g(x) g(x) g(x) 0 x    1 ; 0  ;   2 g(x) 0 x 0; 0216 15 g 1I g g(x)  x² 1 ln x 0;  g x 2 g '(x) 12 g '(x)  2x  1  2x² 1 x  2 22 xx x  2x² 1  0 g'(x)  0 g '(x) x  1 0 g x 0; g(x) 0 g( 2 ) 2 2 g( 2 )   1 2 1 ln 1  3  1 ln 2 2  2  2 22 g( 2 ) 0 g 2 g(x) 0 x 0; limf (x) lim f (x) 1 II x x 0 12 0202

f (x)  ln x  x 1 0; f x lim 1   lim ln x   lim 1 .ln x   xx 0 x 0 xx 0 lim ln x  0 limf (x)  lim(x 1)   xx x x f '(x)  g(x) x 0; 0 x² f '(x)   ln x  ' 1  1 .x 1ln x 1 x² 1 ln x  g(x)  x  x x² x² x² f f f '(x)  g(x) x² x 0 f '(x) f  f (x)   (T ) 3 y  f '(x )(x  x )  f (x ) (T ) 0 00 y  2x  2 y  f '(1)(x 1)  f (1) y  x 1 () (C) 4 limf (x)  y  0 (C) y  x 1 x lim ln x  0 limf (x)  y  lim ln x  0 xx x xx () (C) f (x)  y  ln x () (C)  x ln x x0 1 f (x)  y 0 () (C) () (C) () (C) 11 0202

(C) () (T ) 5 u(1;2) A(1;0) (T ) v(1;1) A(1;0) () y () (T) 3 2 1 0 1234567 x -1 -2 (C) ( ) A(1;0) 6 m y  mx  m (1;0) A(1;0) ( ) ( ) m y  m(x 1) y  mx  m m A(1;0) f (x)  mx  m y  f (x) f (x)  mx  m y  mx  m f (x)  mx  m ( ) (C) A(1;0) m m1 1 m2 3 1 m 22 m 24 0216 16 g 1I g(x)  1 (x 1)e  2ln(x 1) 1;  g  1 lim 2ln(x 1)   lim (x 1)e  0 lim g(x)   x 1 x 1 x 1 limln(x 1)   lim(x 1)e   limg(x)   x x x g g '(x) 10 0202

1;  g x 1; g'(x) 0 g '(x)  e  2 x x 1 g '(x) g 1  g(x)   0,34  0,33  g(x)  0 0 limg(x)   lim g(x)   1; g x x 1 g(0,33)  0,02 g(0,34)  0,03 0,34  0,33 g()  0  3 1; g(x) g(x) g(x) 0 x ; g(x) 0 x 1; limf (x) lim f (x)   1 II x x 1 f f (x)  e  ln(x 1) 1;  x 1 (x 1)² lim e(x 1)  ln(x 1)  lim 1 .ln(x 1)  ()   lim f (x)   x 1 (x 1)² x 1 (x 1)² x 1 x  1 (C ) lim f (x)   f x 1 lim ln(x 1)  0 lim 1  0 lim 1 e  ln(x 1)   0 limf (x)  0 x (x  1) x x  1 x  1 (x 1)  x x y0 (C ) limf (x)  0 f x f '(x)  g(x) : 1; x (x 1)3  ln(x 1)  1 (x 1)²  2(x 1)ln(x 1)  e  '  (x 1)²  e (x  1) f '(x)     '    1)²  x 1 (x (x 1)4     e(x 1)  1  2ln(x  1)     1  e(x 1)  2 ln(x  1)    g(x)  (x 1)3 (x 1)3   (x 1)3  (x 1)3 1;  f 13 0202

g(x) f '(x) f '(x)  g(x) (x 1)3  0 f f () x 1  f '(x)  f (x) 0 (Cf ) y 3 2 1 -1 0 1 2 3 4 5 6 (7Cf ) x -1 -2 k(x)  f ( x ) 1;1 k3 k(x)  k(x) x 1;1 x 1;1 k k k x  x k(x)  f ( x )  f ( x )  k(x) (C ) (C ) (C ) k f k y k(x) 4 ((CCk )) k(x)  f (x); 1 x  0 3k f (x);0 x 1  2 (C ) (C ) x 1;0 fk (C ) (C ) x 0,1 : 1 fk k -1 0 1x k(x)  m -1 y  k(x) k(x)  m y  m -2 14 0202

k(x)  m ( ) (C ) m k me 2 (C ) k m e1 e m k() 3 m  k() 4 m k() 5 0215 17 () () 1I () () () () x 0;  () () x () () x ; x g(x) 0 g(x) g(x)  ln x  (x  3) g(x) x 0; g(x) x   g(x) x ; 2,2  2;3 3 g '(x)  1 1 0 0;  g x g(2;3)  0;13 g(2;2)  0;01 2,2  2;3 limf (x) lim f (x) 1 II x x 0 lim f (x)  (ln x  2)   lim f (x)  (1 1)   lim f (x)   x 0 x 0 x x 0 lim(ln x  2)   lim(1 1 )  1 limf (x)   x x x x . f '(x)  g(x) x 0; 0 x² f '(x)  ( 1 )(ln x  2)  (1 1)( 1)  (ln x  2)  (x 1)  g(x) x² x x x² x² f 15 0202

x 0   f '(x) f '(x)  g(x) x² f '(x) -0 +   g(x) f (x) f f () f ()  (1 )² 3 f ()  ln   3   g()  0 f ()  (1 1 )(ln   2)  f ()  (1 1 )(3    2)  ( 1)(1 )  ( 1)²   (1,2)² ( 1)² (1;3)²....(2) 1,2 ( 1) 1;3 2,2  2;3......(1) 0;73 f () 0;66 (1, 2)² ( 1)² (1;3)² (2) (1) 2;3  2;2 (C ) 4 f f (x) (C ) f (x 1)  0 (ln x  2)  0 (1 1)(ln x  2)  0 f (x)  0 x f (x) x  1 (x 1)  0 x  e² (ln x  2)  0 x 0 1 e²  lnx-2 x 0;1  e²; x-1 (C ) f(x) f x  e² x 1 x  e² x 1 (C ) f y (C ) x 1;e² 4 (Cf ) f 0; e² (C ) f 3 2 1 0 1 2 3 4 5 6 7 8x 16 0202

0214 18 limf (x) lim f (x) 1 x x 0 lim 1   lim ln x   lim f (x)  1  2 lim ln x   xx 0 x 0 xx 0 x 0 lim ln x  0 limf (x)  1  2lim ln x  1 xx x xx x0 (C ) lim f (x)   y 1 f (C ) x 0 f f limf (x)  1 x 0;  (1 .x  1ln x) (1  ln x) 0;  x f '(x)  2  2 xe f x² x² xe f '(x)  0 x0 e  1 ln x  0 f '(x) 0 1 ln x 0 f'(x) 0 f (e) x e 1 ln x 0 f '(x) 0 f f (x)  1 y 1 () (C ) 0 f y  1 f (x)  y  2 ln x x (1;1) () (C ) x  1 ln x  0 f (x)  y  0 f () (C ) x 0;1 ln x 0 f (x)  y 0 f () (C ) x 1; ln x 0 f(x)  y 0 f 1 A (Cf ) (T) y  2x 1 y  2(x 1) 1 y  f '(1)(x 1)  f (1) (T) 01  f (x)  0  f(e0,3)=0,2 f (e0,4)  0,2 f (e0,4 )  f(e0,3) 0 0 ;e f g(α)  0 α  e0,4 ;e0,3  17 0202 (C ) (T) 3 f

h(x)  h(x)  0 x -4 h(x)  h(x)  1 2 ln x 1 2 ln x  2 ln x  2 ln x  0 x  * x x x x h h(x)  h(x) h(x)  h(x)  0 (C ) (C ) f h x 0; (C )  (C ) h(x)  f (x) x ;0 hf (C ) (C ) h(x)  f (x) fh (C ) (C ) (C ) (T) f hf (C ) y f 2 (C ) 1 (C ) h h -5 -4 -3 -2 -1 0 1 2 3 4 5x -1 -2 -3 m  h(x) ln x²  (m 1) x x ln x²  (m 1) x y  m ln x²  (m 1) x y  h(x) (C ) ( ) : y=m m h 1 m h(e) 2 m  1 (1 m h(e) 4 m  h(e) 3 18 0202

0213 19 g 1I x 1; g(x)  x²  2x  4  2ln(x 1) lim g(x)  lim (x²  2ln(x 1)) x x lim g(x)  lim (x1)( x²  2 ln(x 1))   x x (x 1) (x 1) lim g(x)  lim (2ln(x 1))  2()   x 1 x 1 g'(x)  2(x 1)  2  2(x 1)² 1  2x(x 2) x 1; (x 1) (x 1) x 1 x 1 0 1;  x g '(x) g '(x) 0 g(x) 0  0 x 1; g(x)   g(x)  4 4g g(x) 0 x 1; 4 lim f (x) 1 II x 1 x 1; f (x)  x  1  2ln(x 1) x 1 (C ) x  1 lim f (x)  1 1 2ln(x 1)   f x 1 x 1 limf (x) x limf (x)  lim x  1  2 ln(x 1)   x x x  1 x 1 f '(x)  g(x) x 1; 2 (x  1)² 2 (x 1) 1(1 2ln(x1))  (x 1)²  3  2ln(x1))  g(x) f '(x)  1 (x 1) (x 1)² (x 1)² (x 1)² f 1;  f 19 0202

f '(x) 0 g(x) f '(x) x 1 0   α f(x)  0 f '(x) f f (x)   lim f (x)   lim f (x)   1; x 1 x -1 0 f (α)  0 α  1;  0  0,5 0  0,5 f (0) 0 f (0,5) f (0,5)  0,37 f (0)  1  (C ) y  x () 3 f lim(f (x)  x)  0  (C ) y  x () f x lim(f (x)  x)  lim 1  2 ln(x 1)  0 x x x  1 x 1 () (C ) f f (x)  x   1 2ln(x 1) x 1 x  e 1 1 2ln(x 1)  0 f (x)  x  0 x  e 1; f (x)  x 0 x 1; e 1 f (x)  x 0 e 1 () (C ) f  e 1; () (C ) 1; e 1 () (C ) f f (T) (C ) x0 4 f f '(x )  1 (C ) (T) y  x  2 0f e3 x  e e 1 g(x )  (x 1)² g(x ) 1 f '(x )  1 0 00 0 0 (C ) (T) f (x  1)²0 f(x)  x  m m 1 ( ) yxm y  x  m f(x)  x  m m m 2 2 y  f (x) y  f(x) ( )  (T) m e3 (C ) m0 1 f ( )  () m 02 0202

0m 2 f(x)  x  m e3 y 3 (Δ) 2 1 -2 -1 0 12345 x (T) (C ) f -1 -2 -3 0210 02 lim f (x) 1 x 0 lim 6 ln  x x 1 )   lim (x  5)  5 lim f (x)     x  0 x 0 x 0 x0 (C ) f lim f (x) x lim ln  x x 1   0 lim (x  5)   lim f (x )  lim (x  5  6 ln  x x 1          x x x x f '(x) x² x 6 x ;0 0 x(x 1) f '(x) 1 6( 1 1 ) x² x 6 f (x)  x  5  6((ln x  ln (x 1) ) x x 1 x(x 1) f x  x3 x 2 x² x 6 0 f '(x) x² x 6 0 0 x(x 1) f '(x) 2 f '(x) 0 f 01 0202

x  2 0 f ( 2) 3 6ln(2) 6ln 3 0,56 0 () 3 f '(x) f(-2)  (C ) yx5 f (x) f lim(f (x)  y)  6limln x  0 x x x  1    (C ) () f () (C ) f x  1 x ;0 (f (x)  y)  6ln x x 1 x 1 () (C ) (f (x)  y) 0 ln x  0 f x 1  f(x)  0 4 f (1)  0,15 f (1,1)  0,02   2;0 f f ()  0  f (3,4)  0,15‫ و‬f (3,5) 1,33 ‫ و‬  ;2 f f ()  0  y () (C )f 5 5 y  1 x  7  6ln 3 6 22 4 () (AB) 4 3 y  1 x  7  6ln 3  3  6ln 3 A 2A 2 4 4 2 y  1 x  7  6ln 3  5  6ln 3 B 2B 2 42 4 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x 1 (AB) y  1 x  7  6ln 3 22 4 -1 M (C ) (AB) 0f (Cf ) -2 x f '(x )  1 (C ) (AB) 20 0 f M 0 -3 x² x 6 1 f '(x )  1 00 02 x (x 1) 2 00 x  4 x  3 x2  x 12  0 2x2  2x 12  x2  x 00 00 00 00 M (3;f (3)) f (3)  2  6ln 3 04 00 0202

0211 01 x  1  I g'(x) 1  g g(x)  1 g(x) 0 x ;1  1; g(x) 0 0 g(x) 1 x x 1; 0 g(x) 1 lim f (x) lim f (x) 1 II x x 1 lim f (x)  lim x  1  lim ln  x  1   1  0  1 lim f (x)  lim x  1  lim ln  x  1   0     x  1  x  1  x  1  x  1  x x x x 1 x 1 x 1 y 1 (Cf ) x  1 (Cf ) x 1; g '(x)  2 0 (x 1)² g '(x)  1(x 1) 1(x 1)  2 (x 1)² (x 1)² f '(x) f '(x)  g'(x)  1  1  2  2 0 f (x)  g(x)  ln(x 1)  ln(x 1) x 1 x 1 (x 1)² (x 1)(x 1) x1 x 1; f f'(x) f  f(x) 1 f (x)  0  1;  f f (3,63) 0 f (3,62) 0 x x 3,62 ;3,63 0 y0 (C ) f 1 f (x )  0 0 0 1 2 3 4 5 6 7 8 9 10 x -1 -2 -3 03 0202

1;  ln  x  1  3  x  1  ln  x  1102 120 lng(x) ln1 g(x) 1 I  x  00 I f 1I lim f (x)  1  limln(2x 1)   lim f (x)  1 lim ln(2x 1)   x 1 x 1 x x 22 fI f 0 f '(x) 0 x 0,5  x1 02 f'(x) 2 2x 1 f(x) f '(x) 0  I f  (d) (C ) 3 f (d) 3 f '(x ) 1 4 x 0 02 f (x) ln(x a) b x I b 1 ln 2 a = - 1 f (x) 1 ln(2x 1) 1 ln 2(x 1) 1 ln 2 ln(x 1) 2 22 ln (C) (C ) f V(1 ;1 ln 2) (C) (C ) f (x) 1 ln 2 ln(x 1) 2 f 2 lim x   1 x 2x 1 2 (C) (C ) f lim g(x)   lim g(x) 1 II x x  1 2 lim g(x)  lim (f (x)  x) lim g(x)  lim (f (x)  x)   x x x 1 x 1 22 lim ln(2x 1)  0 lim g(x)  1  lim (2x  1)  x 1  ln(2x 1)    x 2x 1 2x  2x 1  x x g0 2x 3 g '(x) g '(x) f '(x) 1 2 1 2x 3 2x 1 2x 1 04 0202

x1 3 g  g( 3) 1 ln(2) 3 0,9 2 2 g'(x) 0 22 g(x) g( 3) 2   g(x) 0 α   3 ; +  g(1) 3  2  g(1) = f(1) -1 = 1+ ln1-1 = 0 lim g(x)   g( 3) 0, 9  3 ; +  2  2  x α   3 ; +   2  g(3)  f (3)  3  2  ln5  0,39 g(2)  f (2)  2  1 ln 3  0,38 2α3 2 α 3 g(3) 0 g(2) y (C)  1 ; 5 (C )  2 g 3 (C ) f I g(x) 4 2 1g g(x) (C )0 1 2 3 4 5 6 7 x x1 1 α  g 0+ 0 -1 2 - g (x) -2 (d) (C ) f g(x) f (x)  y  g(x) (d) (C ) f (d) (C ) x 1;α (d) (C ) x 0,5;1  α; f f (d) (C ) x  α x  1 f f (x) 1;α x 1;α 5 f (α)  α f (1)  1 g(1)  g(α)  0 1 f (x) α f (1) f (x) f (α) 1; α f 05 0202

0229 03 lim h(x) lim h(x) 1 x x 1 1  h(x)  x²  2x  ln(x 1) lim h(x)  lim (1 2  ln(x 1))   lim h(x)  lim (x²  2x  ln(x 1))   x x x 1 x 1 h '(x)  1 2(x 1)² 0 x 1 h '(x)  2x  2  1  2(x 1)² 1 x 1 x 1 x1 0 h h h '(x) 0 x 1 h'(x) h(x) h(0) 3 h(x)  h(x) h(0)  0 0  x 1 0  h(x) 0 t   u   lim ln u  0 ux u  et t  ln u lim ln u  lim t  lim 1 0 u et et x x x t lim f (x) x lim f (x)  lim (x 1)  lim ln(x 1)    0   x x x x  1 lim [f (x)  (x 1)] x lim [f (x)  (x 1)]  lim ( ln(x 1))  0 x x x 1 (Cf )  y  x 1 (Cf ) lim [f (x)  (x 1)]  0 x (Cf ) 06 0202

[f (x)  (x 1)]   ln(x 1) x 1 x+1 0 ln(x 1) (0; 1) (Cf ) x  0 ln(x 1)  0 ln(x 1)  0 (Cf ) 1 x  0 ln(x 1)  0 ln(x 1)  0 (Cf ) x  0 ln(x 1)  0 ln(x 1)  0 f '(x)  h(x) 0 (x 1)² x 1 (x  1)  1ln(x  1) (x  1)² 1 ln(x  1) x²  2x  ln(x  1) h(x) 1 f '(x)  1    (x 1)² (x 1)² (x 1)² (x 1)² h(x) f '(x) x1 0  y2 (Cf ) 3 f'(x) 0  f(x)  2 y2 (Cf )  f(0)  3, 4 3, 3  f(x) 3,3 3,4 f f (3,4)  2,06 f (3,3)  1,96 f (3,3) 2 f (3,4) f ()  2 (Cf ) 4 (C ) y f y  x 1 x 07 0202

08 0202

0219 04 0;  g(x) lim g(x) I g x 0 0;  g(x)  (x 1)(x  e)  e(x ln x) lim (x 1)(x  e)  e lim x ln x  0 lim g(x)  e x 0 x 0 x 0 x 0; g(x)  0 lim g(x)  e 0; g x 0 lim f (x)   lim f (x) 1 II x x 0 f (x)  ln(x 1)  eln x 0;  f x 1 lim ln x   lim ln(x 1)  0 lim f (x)   x 0 x 0 x 0 lim ln(x 1)  lim ex . ln x    e(0)   lim f (x)   x x x 1 x x f '(x)  g(x) 0; x x(x 1)² f '(x)  1  e ln x  e  (x 1)(x  e)  ex ln x  g(x) x 1 (x 1)² x(x 1) x(x 1)² x(x 1)² f g(x) f '(x) f '(x)  g(x) x(x 1)² x0 f f '(x)  f (x)   0 1 (C ) (T) 3 y  e 1 x  e 1  ln 2 f 22 A y  f '(1)(x 1)  f (1) (T)  f (x)  0 09 0202

0;  ff f ()  0 lim f (x)   lim f (x)   x x 0 0;   A 0,7    0,8 0,7    0,8 f (0,7)  0  f (0,8) lim f (x)  ln(x 1) 4 x lim f (x)  ln(x 1)  lim ex . ln x  e.0  0 x x x  1 x  (C ) (T) lim f (x)  ln(x 1)  0 ln x f x (C ) () f f (x)  ln(x 1)  e ln x x 1 () (C ) 0 x 1 f (1;ln 2) () (C ) x 1 f () (C ) x 1 f (C ) () (T) f y 2 1 -1 0 1 2 3 4 5 6 7x -1 -2 -3 v(1 ) x  ln x () 0 0218 05 0;1 g 1I 0;1 g(x)  2  x  ln x g 32 0202

g '(x)  1 1  x 1 0;1 g xx g g'(x)  0 x 0;1 0,15    0,16  g(x)  0 0 g(0,16)  g(0,15)  0;1 g 0,15    0,16 g()  0  0;1 g(x) 3 g(x) g(x) 0 x ;1 g(x) 0 x 0; lim f (x) lim f (x) 1 II x x 1 f (x)  1 2x  ln x 1;  f x 1 lim (x 1)  0 lim (1 2x  ln x)  1 lim f (x)   x 1 x 1 x 1 lim f (x)  lim 1 2x  lim x . ln x  2 1.0  2 x x x 1 x x 1 x (Cf ) x  1 lim f (x)   (Cf ) y  2 x 1 g( 1 ) f '(x)  x x lim f (x)  2 (x 1)² x 0 (2  1 )(x 1) 1(1 2x  ln x) 2  1  ln x 2  1  ln 1 g( 1) x x  x x x f '(x)  (x 1)² (x 1)² (x 1)² (x 1)²  1 ;  1; 1  f     g( 1 ) f '(x) f '(x) x g(1)  0 x  1; 1  g(x)  0 x ;1 x   g(1)  0 x   1 ;  g(x)  0 x 0; x   31 0202

g  1 ;   1; 1  f x       f '(x) 1 1 2 3  0 1 ln x  0 1 1 ln x  0 2 f (x) f ( 1 ) 1 ln x  0 3  4  () (C ) y  2 f f (x)  y  1 2x  ln x  2  1 ln x xe x 1 x 1 xe 1 ln x xe (e; 2) () (C ) () f e;  () 0;e (C ) f 1y (C ) f (Cf ) 0 1 2 3 4 5 6 7 8x -1 -2 -3 -4 -5 f (x)  m m 5 ym f (x)  m (C ) y  f (x) f f (x)  m f (x)  m 2  m  f ( 1 )  f ( 1 )  m  2  30 0202

0217 06 lim f (x) lim f (x) 1 x 1 x 2 f (x)  2x  3  2 ln  x 1  ;1  2; f  x 2  lim 2x  3 1 lim ln  x  1    lim f (x)    x  2  x 1 x 1 x 1 lim (2x  3)  1 lim ln  x  1    lim f (x)    x  2  x 2 x 2 x 2 x 1 (C ) lim f (x)   x2 f x 1 (C ) lim f (x)   f x 2 limf (x) limf (x) x x lim2x  3   lim ln  x  1   ln1  0 limf (x)    x  2  x x x lim2x  3   lim ln  x  1   ln 1  0 limf (x)    x  2  x x x f '(x)  2  2 : D x 0 (x 1)(x  2) f (x)  2x  3  2ln(x1)  ln(x 2) f (x)  2x  3  2 ln  x 1   x 2  f '(x)  2  2  1  1   2  (x 2  2)  x1 x 2  1)(x f f '(x) f '(x)  2   (x 1  2)  0 1 1)(x  f x  f f '(x) 1 2  f (x)     3 f (3  x)  f (x)  0 (3  x)  D D x xD ff f (3  x) D (x) ;2  1; f 33 0202

f (3  x)  f (x)  2(3  x)  3  2 ln (2  x)   2x  3  2 ln (x  1)  (1  x)  (x  2)   2 ln (x 1)(2  x)   2 ln (x 1)(x  2)   2 ln1  0 (x  2)(1 x)  (x  2)(x 1)  (C ) f f (2. 3  x)  f (x)  2(0) f (3  x)  f (x)  0 2 (3 ;0) (C ) 2 f 0,45 ;0,46  f (x)  0 4 5 limf (x)   lim f (x)   ;1 f x x 1 f (0,46)  0,015 0 f (0,45)  0.027 0  0,45 ;0,46 f ()  0   f (x)  0 f (x)  0  (C ) f     2(3) 2;   2 (;0)  (;0) 3 3 2,54 ;2,55 3  0,46 ;3  0,45  0,45 ;0,46 (C ) y  2x  3 () f (C ) y  2x  3 () limf (x)  y  0 f x  lim f (x)  y  lim 2 ln  x 1   ln1  0  x 2  x  x  () (C ) f f (x)  y  2 ln  x  1  () (C )  x  2  f x 1 1 x ;1 f (x)  y 0 x2 x 2; f (x)  y 0 x 1 1 x2 34 0202

x  1 2  f (x)  y () (C ) f () (C ) (C) f () y (C ) f 4 () 3 2 1 -3 -2 -1 0 1 2 3 4 5x -1 -2 -3 0217 07 limg(x) lim g(x) 1I x x 0 g(x)   1  2  ln x 0;  g 2 x² lim x²  0 lim 2  ln x   lim g(x)   x 0 x 0 x 0 lim ln x  0 lim 2  ln x  lim 2  lim ln x  0 limg(x)   1 x ²x  x ²x  x² x²x x 2 x g g g '(x)  1 (x²)  2x(2  ln x) 5  2 ln x x g '(x)   x4 x3 5  2ln x g '(x) 35 0202

x  e² e; 5 5  2ln x  0 5  2ln x 0 x  e2  e² e 5  2ln x 0 x  0;e² e  x0 e² e  0 g '(x) 0 1  g(e² e) 2 g(x)  g(x)  0 1,71  1,72 g(e² e) 0 lim g(x)   x  0;e² e  g x 0 g(1,71)  g(1,72) 0 g(1,72)  0,07 g(1,71) 1,31 1,71  1,72 g()  0  g(x) g(x) 0 x ; g(x) x 0; g(x) 0 limf (x) lim f (x) 1 II x x 0 f (x)   1 x  2  1 ln x 0;  f 2x lim ( 1 x  2)  2 lim 1 ln x   lim f (x)   2x 0 x 0 x x 0 lim( 1 x  2)   lim 1 ln x  0 limf (x)   2x x x x f0 f '(x)   1  ( 1 )(x) 1(1 ln x)  1  2  ln x  g(x) x 2 x² 2 x² f g(x) f '(x) x 0   f '(x) 0 f (x) f ()   36 0202

.(C ) y1x2 () 0 f 2 limf (x)  y  0 (C ) y1x2 () f2 x limf (x)  y  lim 1 ln x  0 f (x)  y  1 ln x x x x x () (C ) f () (C ) f 1 ln x f (x)  y (e;f (e)) () (C ) xe 1 ln x  0 4,19  4,22 f 1 ln x 0 1 ln x 0 () (C ) x 0;e f 3 () (C ) x e; f (C ) () f 0,76  0,77 f ()  f ()  0  (C ) f y (C ) () 3 f 2 () 1 0 1234567 x -1 -2 (C ) f -3 1 0216 08 limg(x) lim g(x) 1I x x 1 g(x)  x 1  ln(x 1) 1;  g x 1 37 0202

lim ln(x 1)   lim x 1   lim g(x)   x 1 x  1x 1 x 1 limln(x 1)   lim x 1  1 x x x  1 limg(x)   x 1;  g g g '(x) g '(x)  ( x 1)' (ln(x 1))'  2  1  x  3 x 1 (x 1)² x 1 (x 1)² x 1 g'(x) 0 x 1  g '(x)  g(x)  0,4  0,5  g(x)  0 0 limg(x)   lim g(x)   1; g x x 1 g(0,4)   3  ln1.4 g(0,5)  g(0,4) 0 g(0,5)   1  ln1.5 7 3 0,4  0,5 g()  0  1; g(x) g(x) 0 x ; g(x) x 1; g(x) 0 lim f (x) limf (x) 1 II x 1 x f f (x) 1 (x 1)ln(x 1) 1; lim (x 1)  2 lim ln(x 1)   lim f (x)   x 1 x 1 x 1 x  1 (C ) lim f (x)   f x 1 lim(x 1)ln(x 1)  lim(x)ln(x)   limf (x)   x x x 1; f 0 f '(x)  (x 1)'.(ln(x 1))'  1.ln(x 1)  x 1  g(x) x 1 f g(x) f '(x) 38 0202

x 1   f '(x) 0 f (x)   f () f () f ()    4  4  1   1  ln( 1) g()  0 f () 1 ( 1)ln( 1)  1 f ()  1 ( 1) ( 1)  ( 1)  ( 1)²  ²  3    4  4 ( 1) ( 1) ( 1) ( 1) 3,5   4 3,6.....(1) 0,4  0,5 4 4 4 ...(2) 1,4  1 1,5 0,4  0,5 1,5  1 1,4 0,64 f () 0,93 (1) (2) h '(x)  f '(x)  f '(a) 1; x 3 f '(a)  0 (x  a)' 1 h '(x)  f '(x) f '(a)(x  a)  f (a)'  f '(x)  f '(a) g h '(x) g xa h '(x)  f '(x)  f '(a)  g(x)  g(a)  0 g xa h '(x)  f '(x)  f '(a)  g(x)  g(a)  0 h '(x) x 1 a  h '(x) 0 h h(x)  0 h x a; h '(x) h x 1;a a h(a)  f (a)  f (a)  0 0h (T ) (C) (T ) a a y  f '(a)(x  a)  f (a) f (x)  y  h(x) (T ) (C) a a (T ) (C) h(x)  0 a (T ) A(1;0) (T ) a 4 a y  f '(a)(x  a)  f (a) 39 0202

0  f '(a)(1 a)  f (a) A(1;0) (T ) 0  g(a)(1 a)  f (a) a 0  f '(a)(1 a)  f (a) a  3 a  0 a²  3a  0 (a 1)²  a 1 (1 a)(a 1) 1  0 (a 1) (3;2ln 4) (0;1) (C) (T ) y  x 1 a y  f '(0)(x  0)  f (0) (T ) 0 y   1  ln 4  x  1  ln 4 y  f '(3)(x  3)  f (3) (T )  2  2 3 (C) y 5 (T ) 4 (C) 0 3 2 1 (T ) 3 -2 -1 0 1 2 3 4 5 6x 0 0216 09 lim g(x) lim g(x) 1I x x 0 lim x ln x  0 lim g(x)  lim (x  x ln x)  0 x 0 x 0 x 0 lim g(x)  lim(x  x ln x) x x lim g(x)  lim x(1 ln x)   x x 0;  g g'(x) 1 (1.ln x  x. 1)  ln x x  0;  x ln x g '(x) 42 0202

x0 1  g '(x) 0 x 0 g g '(x) 1  0 g(x) 1 0   2 3,5  3,6 g(x)  1 limg(x)   lim g(x) 1 1; g x x 1 g(3,6) 1 g(3,5) g(3,6)  1,01 g(3,5)  0,88 3,5  3,6 g()  1  0;  g(x) 1 3 g(x) 1 g()  1 g(x) 1  0 g(x) 1 0 x ; g(x) 1 0 x 0; y0 x0 (C ) 1 II (C ) f x0 f lim f (x)  lim ln x  lim ln x   x 0 x  1x 0 x 0 ln x y0 (C ) limf (x)  lim ln x  lim x  0 f x  1 1  1x x x x f '(x)  g(x) 1 0;  x2 x(x 1)² 0;  x f (x)  ln x ;  x 1 f '(x)  1 (x 1) 1.ln x  x  x ln x 1  g(x) 1 x (x 1)² x(x 1)² x(x 1)² 0;   f g(x) 1 f '(x) f '(x)  g(x) 1 f x(x 1)² 0;   g(x) 1 0 x 0; 41 0202

;  f g(x) 1 0 x ; x 0 f f '(x)    f (x) 0 1 f () 0 (T) (C ) (T) f y  f '(a)(x  a)  f (a) y  1 x  1 y  f '(1)(x 1)  f (1) 22 0;  lim f (x)  f () x x   f  (C ) lim f (x)  f ()  f '()  0 y  f () f x x    f ()  1 3  ln   1    ln  1 g()  1  f ()  ln  ...(1)  1 f ()   1  1 1 ( 1)  ln  102 f () 1,25 ln  1,28....(2) 4,5  1 4,6...(1) 3,5  3,6 f ()  ln   1 0,27 f () 0,28 21 f (x)  1 x  m (C ) 4 2 f x(x 1)  2m(x 1)  2ln x (E) x²  x  2m(x 1)  ln(x²)..(E) f (x)  1 x  m 1 x  m  ln x 1 (x 1) 2 2 x 1 2 40 0202

(E) m (C ) y  1 x  m (E) f2 h m1 2 5 h(x)  h(x) x  * x  * h x  x h(x)  ln x ln x h   h(x)  x 1  x 1 x * (C ) (C ) (C ) (C )  hf f h (C ) (C ) x * f h  y (C ...) h 1 (C ) ‫ـــ‬ f -5 -4 -3 -2 -1 0 1 2 3 4 5x -1 0215 32 limh(x) lim h(x) 1I x x 2 2;  h(x)  (x  2)²  2  2hn(x  2) lim (2ln(x  2))   lim h(x)  lim (x  2)²  2  2ln(x  2)   x 2 x 2 x 2 lim h(x)  lim(x  2)²  2  2ln(x  2) x x lim h(x)  lim(x  2) (x  2)  x 2 2  2 ln(x  2)     (x  2)  x x 43 0202

lim  x 2 2   0 lim  ln(x  2)   0 lim(x  2)       (x  2)  x x x h 0 D  2; h(x)  (x  2)²  2  2hn(x  2) h h '(x)  2(x  2)  2  2(x  2)²  2  2(x  2) 1(x  2) 1 x2 x2 (x  2) h '(x)  2(x 1)(x  3) (x  2) x  3 D x  1 (x 1)(x  3)  0 h '(x)  0 h (x 1)(x  3) 0 h '(x) 0 x 2;1 x 1; (x 1)(x  3) 0 h '(x) 0 x 2 h h '(x)  -1  0  h(x) h(1) h(x)  0 2; x 3 h(x) 0 h(1)  3 0 h limf (x) lim f (x) 1 II x x  2 2;  f (x)  x 1  2 ln(x  2) x2 lim ln(x  2)   lim 2   lim f (x)   x 2 x  2x 2 x 2 x  2 (C ) lim f (x)   f x 2 lim(x 1)   lim 2ln(x  2)  0 limf (x)   x x x  2 x f '(x)  h(x) 2; x 0 (x  2)² f '(x)  1 2 ln(x  2)  2  1  (x  2)²  2ln(x  2)  2  h(x) (x  2)² (x  2) (x  2) (x  2)² (x  2)² 2; f 0 44 0202