Limites
Exercice 1:
1. Étudier les limites suivantes :
1. limx→−∞(x3− 2x + 5) 2. limx→−∞(x4+ 2x) 3. limx→+∞(x3− x4) 4. limx→−1(x3− 5x + 1) 5. limx→+∞(x−32+5) 6. limx→0( x 2 x+2) 7. limx→1( x 2 −1 x2+3x+5) 8. limx→0(x 3 −1 x2 ) 9. limx→2( x 2 +5 (x−2)2) 10. limx→1(x2−2x+13−x ) 11. limx→1(3xx+1−2) 12. limx→+∞( 2x 2 +5 x2−3x+1) 13. limx→+∞(3x 2+5x 1−3x2 ) 14. limx→+∞(2xx+32+5) 15. limx→−∞(x3+4x+13−x ) 16. limx→−∞(x 4+3x x+3 ) 17. limx→+∞(x 2+2x+1 x ) 18. limx→+∞( x 3 x2+3x+5)
19. limx→−∞(x 2+1 1−x) 20. limx→0(x 3 −3x x2+x) 21. limx→1( x 2 −x x2+2x−3) 22. limx→−2(x 2+4x+4 x2−x−6) 23. limx→2((x3x+5−2)2) 24. limx→−3((3+x)1−3x2) 25. limx→1(√xx−1−1) 26. limx→4(2− √ x 4−x )
Exercice 2:
Déterminer les limites à gauche et à droite , lorsque x tend vers x0,
des fonctions f dans les cas suivants :
1. xo= 2; f (x) = 3xx−2−7 2. xo= −3 ; f(x) = x+35x 3. xo= −2 ; f(x) = −3x+2x+2 4. xo= 1; f (x) = 13x−x 5. xo= 2; f (x) = 2x2−x−5 6. xo= 0; f (x) = x 2+3x+5 x2+2x 7. xo= −2;f(x) = x 2+3x+5 x2+2x 8. xo= 1; f (x) = (x−1)(x+3)2x 9. xo= −3; f(x) = (x−1)(x+3)2x+1
10. xo= −1; f(x) = (x+1)(23x+7−x)
11. xo= 3; f (x) = 6+xx+2−x2
Exercice 3:
Déterminer les limites suivantes :
1. limx→1x 2 −5x+4 x2−3x+2 2. limx→1 x 3 −3x+2 2x3−3x2+1 3. limx→2 h x+3 x2+3x−10 ³ 8 5(x+2) − 4 x+ 17x+6 5(x2+1) ´i 4. limx→13x−2− √ 4x2−x−2 x2−3x+2 5. limx→0 √ x+1−√x2+x+1 x 6. limx→1 √ x+s−√3x+1 √ x−1 7. limx→2 x− √ x+2 √ 4x+1−3 8. limx→0x 4+1 −√1−x4 x2√1−x4 9. limx→1 √ 3+√2x−1−2 √ 2+√3x+1−√x+3 10. limx→0+ x √ 1+x2−1 11. limx→0− x √ 1+x2−1 12. limx→+∞ (3x 2+1)(5x+3) (2x3−1)(x+4) 13. limx→+∞ (2x−3) 2(4x+7)3 (3x−4)2(5x2+1) 14. limx→+∞ 4x+1+√7x16x2+x+1 15. limx→+∞(3x − √ x2− x + 1)
16. limx→−∞(3x − √ x2− x + 1) 17. limx→+∞ √ x2+x+1+√x2−x+1 x+√x2+1 18. limx→−∞ √ x2+x+1+√x2−x+1 x+√x2+1 19. limx→+∞(2x − 1 − √ 4x2− 4x + 1) 20. limx→−∞(2x − 1 − √ 4x2− 4x + 1)
Réponses
Exercice 1:
1. limx→−∞(x3− 2x + 5) = −∞ 2. limx→−∞(x4+ 2x) = +∞ 3. limx→+∞(x3− x4) = −∞ 4. limx→−1(x3− 5x + 1) = 5 5. limx→+∞(x−32+5) = 0 6. limx→0( x 2 x+2) = 0 7. limx→1( x 2 −1 x2+3x+5) = 0 8. limx→0(x 3 −1 x2 ) = −∞ 9. limx→2( x 2 +5 (x−2)2) = +∞ 10. limx→1(x2−2x+13−x ) = +∞ 11. limx→1(3xx+1−2) = 1 2 12. limx→+∞( 2x 2+5 x2−3x+1) = 2 13. limx→+∞(3x 2 +5x 1−3x2 ) = −1 14. limx→+∞(2xx+32+5) = 015. limx→−∞(x3+4x+13−x ) = 0 16. limx→−∞(x 4+3x x+3 ) = −∞ 17. limx→+∞(x 2+2x+1 x ) = +∞ 18. limx→+∞( x 3 x2+3x+5) = +∞ 19. limx→−∞(x 2+1 1−x) = +∞ 20. limx→0(x 3 −3x x2+x) = −3 21. limx→1( x 2 −x x2+2x−3) = 14 22. limx→−2(x 2+4x+4 x2−x−6) = 0 23. limx→2((x3x+5−2)2) = +∞ 24. limx→−3((3+x)1−3x2) = +∞ 25. limx→1(√xx−1−1) = 2 26. limx→4(2− √ x 4−x ) = 1 4
Exercice 2:
1. limx→2− 3x−7 x−2 = +∞ limx→2+ 3x−7 x−2 = −∞ 2. limx→−3− 5x x+3= +∞ limx→−3+ 5x x+3 = −∞ 3. limx→−2− −3x+2x+2 = −∞ limx→−2+−3x+2x+2 = +∞ 4. limx→1− 3x 1−x = +∞ limx→1+ 3x 1−x = −∞ 5. limx→2− 2x−5 2−x = −∞ limx→2+ 2x−5 2−x = +∞6. limx→0− x2+3x+5 x2+2x = −∞ limx→0+ x2+3x+5 x2+2x = +∞ 7. limx→−2− x2+3x+5 x2+2x = +∞ limx→−2+ x2+3x+5 x2+2x = −∞ 8. limx→1− 2x (x−1)(x+3) = −∞ limx→1+ 2x (x−1)(x+3) = +∞ 9. limx→−3− 2x+1 (x−1)(x+3) = −∞ limx→−3+ 2x+1 (x−1)(x+3) = +∞ 10. limx→−1− 3x+7 (x+1)(2−x) = −∞ limx→−1+ 3x+7 (x+1)(2−x) = +∞ 11. limx→3− x+2 6+x−x2 = +∞ limx→3+ x+2 6+x−x2 = −∞
Exercice 3:
1. limx→1x 2 −5x+4 x2−3x+2 = 3 2. limx→1 x 3 −3x+2 2x3−3x2+1 = 1 3. limx→2 h x+3 x2+3x−10 ³ 8 5(x+2) − 4 x+ 17x+6 5(x2+1) ´i =143 4. limx→13x−2− √ 4x2−x−2 x2−3x+2 =12 5. limx→0 √ x+1−√x2+x+1 x = 0 6. limx→1 √ x+3√−√3x+1 x−1 = 0 7. limx→2 x− √ x+2 √ 4x+1−3 = 9 8 8. limx→0x 4+1 −√1−x4 x2√1−x4 = 0 9. limx→1 √ 3+√2x−1−2 √ 2+√3x+1−√x+3 = −410. limx→0+ x √ 1+x2−1 = +∞ 11. limx→0− x √ 1+x2−1 = −∞ 12. limx→+∞ (3x 2+1)(5x+3) (2x3−1)(x+4) = 0 13. limx→+∞ (2x−3) 2 (4x+7)3 (3x−4)2(5x2+1) = +∞ 14. limx→+∞ 4x+1+√7x16x2+x+1 = 7 8 15. limx→+∞(3x − √ x2− x + 1) = +∞ 16. limx→−∞(3x − √ x2− x + 1) = −∞ 17. limx→+∞ √ x2+x+1+√x2−x+1 x+√x2+1 = 1 18. limx→−∞ √ x2+x+1+√x2−x+1 x+√x2+1 = +∞ 19. limx→+∞(2x − 1 − √ 4x2− 4x + 1) = 0 20. limx→−∞(2x − 1 − √ 4x2− 4x + 1) = −∞
