Sindbad~EG File Manager
ELF > P4 @ @ 1. y = t^3 - 4t^2 + 2 2. y = x^5 - x^3 + 1 3. y = 6x^3 - 7x^5 4. f(x) = x^3 + 3x^2 + 6x 5. f(x) = x^8 - x^6 + x^4 - x^2 + 1 6. y = ax^2 + bx + c 7. y = (1 - 2t) (1 +t) 8. y = x^4 - 3x^3 + 2x^2 - x +1 9. y = 5x^3 - 3x^5 - 7 10. y = 3 (t^2 + t)^2 11. y = (2x^2 + x +1)^10 12. y = x^n 13. y = ((x^3) / 3) + ((x^2) / 2) + x +1 14. y = ((-t)^2) ^5 15. y = 1 / (x - 1) 16. y = 1 / (x^2 - 1) 17. y = 1 / (x - 7) 18. y = (x^2 - 4) / x 19. y = x/(1+ x) 20. y = (x + 2) / (2x + 1) 21. y = (x + 4) / (x^2 + 1) 22. y = (x + 1/x)^3 23. y = (x^2 +1) (x^(- 2) - 1) 24. y = x^3 + x^(- 3) 25. y = sqrt x 26. y = sqrt(3t + 1) 27. y = root (3, x) 28. y = root (4, x) 29. y = root (5, x) 30. y = x^(5/2) 31. f(x) = sqrt x - 10 32. y = x^(1/2) + x^(1/4) 33. y = x^(1/3) - x^(- 1/3) 34. y = x^(3/2) + 2x^(1/2) 35. y = root (3, 3x^2) 36. y = 1 / root (3, 7 - x) 37. f(t) = sqrt (t + sqrt t) 38. y = sqrt (1+x^3) 39. y = root (3, 1 - u^2) 40. y = (1+sqrt t) / (5t) 41. y = (1 - x^2)^(5/2) 42. h(x) =sqrt(x^2 + 1) 43. G(r) = sqrt(r) + root(3, r) 44. y = sin x 45. y = cos x 46. y = tan x 47. y = cot x 48. y = sec x 49. y = csc x 50. y = sin (3x) 51. y = cos (2x) 52. y = sin x + cos x 53. y = 5 tan (x/3) 54. y = cot (2x - 1) 55. y = 1 / (sin x) + 1/ (cos x) 56. y = sin x (cos x) 57. y = cos(3x) 58. y = sin^2 (2x - 1) 59. y = sin^3 (3x) 60. y = cos^4 (4x) 61. y = tan^3(2t) 62. y = x^4 + cot^2 (x) 63. y = (abs x) x 64. y = x^2 (abs x) 65. y = ln x 66. y = e^(x^2) 67. y = e^(x^2 - 1) 68. y = 1 / (e^x - 1) 69. y = ln (e^x - 1) 70. y = x^2 (ln x) 71. y = ln (x^2 + x + 1) 72. y = e^(sin x - 1) 73. y = x^(pi) 74. y = (1-x^2)^(3/4) 75. H(t) = tan^3(2t-1) 76. F(r) = sec(sqrt r) 1. y^3=x^2 2. x^2-3xy=18 3. y^2=x^2+2x 4. x^3 + y^3 = 1 5. x^3 +xy -y^3 = 9 6. x^2 y^3 =1 7. x^2y^2=9 8. (x-1)y^2=x+1 9. x^2 y^3 +y^2 x^3 + x +1=0 10. 1/x + 1/y = 3+x+y 11. x^2 - 6x/y + y^2 = 7 12. y/(x-y) = 2x^3 13. 1/(x+1)+1/(y+1)=1 14. 1/x^2+1/y^2=20 15. (xy+y^3)/(1+x)=1 16. 2x + y - xy^(3/2) = 4 17. x^2+sqrt (xy)=7 18. 2/x + xy^(3/2) = 4(x-y) 19. x^(-1)+sqrt (x-y)= xy 20. sqrt x+(1/sqrt y)=2 21. sqrt x + sqrt y =1 22. (x^2+y^2)^2=4xy 23. (x^2+y^3)^3=8x^2y^2 24. y-1=sqrt(1-sqrt x) 25. x=sqrt ((x^4-x+1)^3)+xy 26. x=e^y 27. e^(xy)= 1 28. e^xy=2x+y 29. x-y^2=e^(y^2) 30. xe^(xy)=2 31. 2e^(x^2 y)=x 32. ln(x-y)=xy 33. ln(x-y/x)=x+y 34. x^4 = x^2 - y^2 35. y+sqrt y ln x=x^2+y^2 36. x sin y = 1 37. x cos y = sin y 38. y^2=x cos y 39. tan xy=x 40. (sin x)(cos y)=y 41. (sin x)(cos y)=x 42. sin x -cos^2 y =x+y 43. x=sin^2(y) 44. y^2 =1-x^2 sec y 45. x^2=sin (y^2) 46. x+y=(sin y) / (cos x) 47. xy = tan(x/y) 48. y^2=sin ((x+5))^(5/4) 49. sin (sqrt y)-3x=2 50. sqrt(xy) + ln(x+y) = 1 1. 3y = 14x^2, diff(x,t) = - 6, x=1 2. A = pi r^2, diff(A,t) = 1 , r = 3 3. y = 4x^2+100, diff(x,t) = 10 , x=10 4. V = x^3, diff(x,t) = 3 , x = 20 5. V = (4/3) pi r^3, diff(V,t) = 100 , r = 50 6. xy = 1, diff(x,t) = 4 , x = 2 7. S = 6x^2, diff(x,t) = 3 , x = 20 8. V = 36 pi h, diff(V,t) = 8 , h = 2 9. W = 100 (1+r / 4000)^(-2), diff(r,t) = 15 , r = 400 10. y = 2x^3 - 3x^2+4, diff(x,t) =2 , x = 2 11. V = 16 x, diff(V,t) = 2 , x =1 12. D = sqrt (y^2+900), diff(y,t) = 15 , y = 30 13. S = u^3, diff(u,t) = -2 , u = 3 14. V = (4/3)pi R^3, diff(V,t) = 20 pi , R = 10 15. V = pi R^2 h / 3, diff(R,t) = 5, diff(h,t) = -6 , R = 4, h = 3 16. y = 3 sqrt (x+9), diff(x,t) = 2 , x=7 17. P = 3 / W, diff(P,t) = 5 , P =9 18. y = sqrt x - 4, diff(y,t) = 3 , x= 4 19. xy = 1, diff(x,t) = 2 , x =1 20. xy = -8, diff(x,t) = 2 , x = 4 21. A = xy / 2, diff(x,t) = 4, diff(y,t) = -3 , x = 20, y = 12 22. A = xy, diff(A,t) = 4, diff(x,t) = 1/2 , A = 800, y=200 23. D = sqrt (16+y^2), diff(D,t) = 3600 , y = 5 24. x^2 + 3xy + y^2 = 1, diff(y,t) = -2 , y= -1, x=0 25. V = (4/3)pi r^3, diff(V,t) = -2, r=3 @ The volume of a sphere is decreasing at 2 cubic meters/second. At what rate is the radius decreasing when the radius is 3? 26. xy^2+4y - 44 = 0, diff(x,t) = 3 , y = 2 27. y = 4x^2 - 100, diff(y,t) = -6 , x = 1 28. 4x^2+y^2 = 25, diff(x,t) = 5 , x = 3/2, y = 4 29. 4x^2+y^2 = 25, diff(x,t) = 5, x = 3/2, y = -4 30. y - x^2+4x = -10, diff(y,t) = 3 , x = -2 31. x^2+xy+y = 16, diff(y,t) = 5 , x = 4 32. u^2 = x^2+(90)^2, diff(x,t) = 20, x = 30, u = -30 sqrt (10) 33. 48 S - 480 = S h, diff(h,t) = 32, S = 30, h = 32 34. y^2 = (30+x)^2 - (18)^2, diff(y,t) = 9, x = 6, y = 18 sqrt 3 35. y = 1 / (x^2 - 1), diff(y,t) = 1, x = -2 36. 2 y^3 - x^2+4x = -10, diff(y,t) = -3, x = -2, y = 1 37. A = 2 h sqrt (16+h^2), diff(A,t) = 2 , h = 3 38. A = x (600 - x) / 2, diff(A,t) = 4 , x = 200 39. x^2+y^3+z^4 = -3, diff(x,t) = -3, diff(z,t) = 4 , x = 2, y = -2, z = 1 40. xyz = 10, diff(x,t) = -2, diff(z,t) = 3, x = 2, y = -5, z = -1 41. V = 4 pi r^3 / 3, S = 4 pi r^2, diff(V,t) = 10 , r = 4 42. x = 20 tan theta, diff(x,t) = 4 , x = 15 43. y - x = sin y, diff(x,t) = 3 , y = pi / 2 44. h = 20 tan theta, diff(h,t) = 3 , theta = pi/6 45. L = 40 sec theta - 20 tan theta+ 30, diff(L,t) = -1 , theta = pi / 4 46. A = ( r theta) / 2 - (sin theta) / 2, diff(r,t) = 1, diff(A,t) = -1 , theta = pi/2, A = 1/4 1. f(x) = (10-x)x, -infinity < x < infinity 2. f(x) = x(x+1),-infinity < x < infinity 3. f(x) = 1+3x, -1 <= x 4. f(x) = 4x+1, x <= 6 5. f(x) = abs(x), -3 <= x <= 2 6. f(x) = abs(3x-1), 0 <= x <= 2 7. f(x) = 1-x^2, 0 < x < 1 8. f(x) = 1-x^2, 0 < x <= 1 9. f(x) = 1-x^2, 0 <= x < 1 10. f(x) = 1-x^2, 0 <= x <= 1 11. f(x) = 1-x^2, -2 <= x <= 1 12. f(x) = 1+(x+1)^2, -3 <= x <4 13. f(t) = 1/t, 0 < t < 1 14. f(t) = 1/t, 0 < t <= 1 15. f(theta) = sin theta, -2pi <= theta <= 2pi 16. f(theta) = tan theta, -pi/4 <= theta < pi/2 17. f(theta) = cos(theta/2), -pi < theta < pi 18. f(theta) = sec theta, -pi/2 < theta <= pi/3 19. f(x) = x^5, -1 <= x <= 1 20. f(x) = 2-x^4, -1 <= x <= 1 21. f(x) = x^2 - 2x+ 2, 0 <= x <= 3 22. f(x) = 1-2x-x^2, -4 <= x <= 1 23. f(x) = x^3-x, -1 <= x <= 1 24. f(x) = 4x^3-15x^2 + 12x + 7, -2 <= x <= 1 25. f(x) = 2x^3 + 3x^2 + 4, -2 <= x <= 1 26. f(x) = 18 x + 15x^2 -4x^3, -3 <= x <= 4 27. f(x) = x^4 - 4x^2 + 2, -3 <=x <=2 28. f(x) = 3x^5 - 5x^3 - 1, -2 <= x <= 2 29. f(x) = x^2 + x/2, 1/2 <= x <= 2 30. f(x) = sqrt(9-x^2), -1 <= x <= 2 31. f(x) = x^(4/5), -32 <= x <= 1 32. f(x) = x/(x+1), 1 <= x <= 2 33. f(x) = abs(x-1)-1, -1 <= x <= 2 34. f(x) = abs(x^2 + x), - 2 <= x <= 1 35. f(x) = x(1-x), x >= 0 36. f(x) = x/(1-x), x >= 2 37. f(x) = sin x + cos x, 0 <= x <= pi/3 38. f(x) = x - 2 cos x, -pi <= x <= pi 39. f(x) = x^3-12x+1, -3<=x <=5 40. f(x) = x^4-x^2,-infinity < x < infinity 41. f(x) = x^5-x^2, -1 <= x <= 1 42. f(x) = 2sin(x)-x, 0 <= x <= pi 43. f(x) = 1/(1+x^2),-infinity < x < infinity 44. f(x) = e^(-x^2),-infinity < x < infinity 45. f(x) = e^sin(x), 0 <= x <= pi 46. f(x) = e^x + e^(-x),-infinity < x < infinity 47. f(x) = 1/x + 1/(1-x), 0 < x < 1 48. f(x) = 2/x + 1/(1-x), 0 < x < 1 49. f(x) = 1/(x^2 + 1) + 1/((x-1)^2 + 1), 0 <= x <= 1 50. f(x) = 1/((x+10)^2 + 1) + 1/((x-10)^2 + 1), -12 <= x <= 12 1. lim(x-> 3, (6 - 2x) / (x - 3)) 2. lim(t-> 0, t (1 - 1/t)) 3. lim(x-> 1, (x^2 - 1) / (1 - x)) 4. lim(x-> -1, (x^2 - 1) / (3x+3)) 5. lim(x-> 2, (x - 2) / (x^3 - 8)) 6. lim(x-> -1, (x^3 - 1) / (x+1)) 7. lim(x-> 2, (3x - 6) / (x^3 - 2x^2)) 8. lim(y-> 1/2, (6y - 3) / (y(1 - 2y))) 9. lim(y-> 1, (1 - y) / (1 - y^3)) 10. lim(x-> -1, ((3+3x) / (x^3+1))^3) 11. lim(t-> -5, (t+5) / (t^2 - 25)) 12. lim(x-> 1, (x^2 - 1) / (x^3 - 1)) 13. lim(x-> 1, (x^5 - 1) / (x^3 - 1)) 14. lim(u-> -1, (u^4 - 1) / (u^3+1)) 15. lim(x-> 1, ((x - 1)^2) / (x^2 - 1)) 16. lim(x-> 2, (x^2+x - 2) / (x^2 - 4)) 17. lim(x-> 2, (x - 2) / (x^2 - 5x+6)) 18. lim(u-> 1, (u^2 - 2u+1) / (u^2 - 1)^2) 19. lim(u-> 2, (u^2 - 7u+10) / (u^6 - 64)) 20. lim(x-> 1, (x - 2) / (x^2 - 10x+25)) 21. lim(x-> 1, (x^2 - x) / (x^3 - x^2+x - 1)) 22. lim(x-> 4, (x^3+4x^2+x+4) / (x^2+3x - 4)) 23. lim(u-> -1, (u^4+1) / (u^3 - 1)) 24. lim(x-> 2, ((x^2 - 4x+3) / (x^2 - 4x+4))^4) 25. lim(t->1/2, (2t^2+5t - 3) / (6t^2 - 7t+2)) 26. lim(u-> 2, (1/u^2 - 1/4) / (u - 2)) 27. lim(x-> 2, (1/x - 1/2) / (x - 2)) 28. lim(x-> 0, (1+1/x) / (1+1/x^2)) 29. lim(x-> 2, (x / (x+1) - 2/3) / (x - 2)) 30. lim(x-> 1+sqrt 2, ((x - 1)^4 - 4) / ((x - 1)^2 - 2)) 31. lim(t-> sqrt 3, (t^4 - 6t^2+9) / (t^2 - 3)) 32. lim(x-> 1, (x^2+x - 2) / (x^5 - 1)) 33. lim(x-> -3, (x+3) / (1/3+1/x)) 34. lim(x-> 1, (1 / (x - 1)) - x^2 / (x - 1)) 35. lim(x-> 1, (x^2+x - 2) / (x - 1)) 36. lim(x->-1, (x^2+6x+5) / (x^2+3x+2)) 37. lim(u-> - 4, (2u / (u+4)) - 8 / (u+4)) 38. lim(y-> infinity, (1 - y) / (1 - y^3)) 39. lim(x-> infinity, ((1 - x)(2 - 3x)) / (10x^2 - 4)) 40. lim(x->infinity,(x - 4)/(x^2-2x - 1)) 41. lim(x-> infinity, (2x^2+x - 1) / (7x^2+3)) 42. lim(t->infinity,(7t^3+3t)/(2t^3+t^2+1)) 43. lim(t-> infinity, t^3 / (t - 3) - t^2) 44. lim(t->-infinity,(7t^2 + 5t)/((1+t)(2t+3))) 45. lim(x-> infinity, (2x^2+x - 1) / (x^3+3)) 46. lim(x->infinity, (x^5-1)/(x^6 - 1)) 47. lim(x-> infinity, (3x^2 - x+5) / ((5 - 4x)(1 - 3x))) 48. lim(x->infinity, (x-2)(x-3)^2 / ((x-4)^2(x+5))) 49. lim(x-> - infinity, ((x+4)(x+1)) / ((2x+7)(x+2))) 50. lim(x->infinity, (x-2)(x-3)^2(x-4)^2 / ((x+5)(x+6)^2(x+7)^2)) 51. lim(x-> infinity, 1/x + 2x / (x+1)) 52. lim(x-> infinity, (x^2+1) / x^4) 53. lim(x-> infinity, (7x - 10) / (x (x+1))) 54. lim(u-> infinity, (u^3 - u - 2) / (2u^3+u^2+1)) 1. lim(x-> -infinity, 5x^(-1/3)) 2. lim(y-> infinity, 1/y^2 + 1/y^4 + 1) 3. lim(x->infinity,sqrt(1+x)-sqrt x) 4. lim(y-> infinity,1/(y sqrt y)) 5. lim(x-> infinity, 3x/ sqrt (4x^2+1)) 6. lim(x-> infinity, (sqrt x) / (4 sqrt x - x)) 7. lim(x-> -infinity, (8+x^2) / (x (x - 1))^(1/3)) 8. lim(x->infinity, (x - sqrt x)/(x + sqrt x)) 9. lim(x-> infinity, x^3 - 1/x^3) 10. lim(x-> infinity, x^2/(x+1) - x^2/(x - 1)) @It would not be correct to replace the denominators by their leading term $x$. That would give zero, which is not the correct answer. This shows that leading terms must be used with care!11. lim(x->infinity,root(3, 1+x)-root(3, x)) 12. lim(x-> infinity, sqrt (x^2-16) / (x - 4)) 13. lim(x-> infinity, (4x - 1) / sqrt (16x^2+1)) 14. lim(x-> infinity, 2x - sqrt (4x^2 - 1)) 15. lim(t-> infinity, t / (t^(1/2) + 2t^(-1/2))) 16. lim(x-> infinity,sqrt((x^2-1)/(x+7x^2))) 17. lim(x-> infinity,1/(5 + sqrt x)) 18. lim(x-> infinity,sqrt(1+x)-sqrt(x-1)) 19. lim(x-> infinity, sqrt(x^2+x -1) + x) 20. lim(x-> -infinity,root(3, x+1) - root(3, x - 1)) 21. lim(x-> infinity, x - sqrt x) 22. lim(x-> infinity, sqrt (x^2/(1+3x^2))) 23. lim(x-> infinity, (x^2 + x) / (x^3+x^2)) 24. lim(x-> infinity, x^3/(x - 5) - x^2) 25. lim(x-> infinity, x - 1/x + x^2 - 1/x^2) 26. lim(x-> infinity, (7x - 10) / sqrt(x (x+1))) 27. lim(x-> -infinity, x / sqrt (4x^2+1)) 28. lim(x-> infinity, sqrt(x) / (x^2 - 1)) 29. lim(x-> infinity, 3 sqrt (x^2/4 - 1) - 3x/2) 30. lim(x-> infinity, sqrt(4 + x^2)/(4+x)) 31. lim(x-> infinity,sqrt(x^2+3x)/(- x+1)) 32. lim(x-> infinity, (1+ sqrt x)/(1- sqrt x)) 33. lim(x-> infinity, sqrt(x^2+x - 1) -x) 34. lim(x-> infinity, sqrt x / (4 sqrt x + 1)) 35. lim(x-> infinity, sqrt (x+50) - x) 36. lim(x-> infinity, sqrt (1+x^2) / x) 37. lim(x-> infinity, x^(3/5) - x^(5/3)) 38. lim(x-> infinity, root(3, x^2+8)/(x+1)) 39. lim(x-> -infinity, root(3, x^3+8)/(x - 1)) 40. lim(u-> infinity, (u^3 - sqrt(u) - 2) / (2u^3+u^2+1)) 41. lim(x-> infinity, 1/x + 5x/(x+1)) 42. lim(x-> infinity, (abs x) / sqrt (x^2+1)) 43. lim(y-> infinity, (9y^3 - y) / (y^3 +1)) 44. lim(x-> infinity, (x+2) / (x+3) - (x^3+2) / (x^3+3)) 45. lim(x-> -infinity, sqrt ((3 - x) / (1 - x))) 46. lim(x-> infinity, x + sqrt(x^2+2x+1)) 47. lim(x-> infinity, sqrt(x^2 - 1)-sqrt(x^2+1)) 48. lim(x-> infinity, (x^(4/3) - x^(1/3) +1) / (3x^(4/3)+3)) 49. lim(t-> infinity, cos(1/t)) 50. lim(x-> infinity, cos^2 x) 51. lim(x-> infinity, 1/ (sec x - 1)) 52. lim(x-> infinity, (1 - cos x) / x^2) 53. lim(x-> infinity, (cos x) / sqrt (x^2 - 1)) 54. lim(x-> infinity, e^(-1/x)) 55. lim(x-> infinity, e^(-x^2)) 56. lim(x-> infinity, ln (- ln (1/x))) 57. lim(x-> infinity, x sin(1/x^2)) 58. lim(x-> infinity, x sqrt (4x^2 -1) / sqrt (9x^3 -1)) 59. lim(x-> infinity, (16 - x^2)^(1/3) / (4 - x)) 1. lim(x->0, 1/x^6) 2. lim(x->0, 1/x^6 - 1/x^4) 3. lim(x->3, 1/(x-3)^8) 4. lim(x-> 0, 1+1/x) 5. lim(x->5+,1/(x-5)) 6. lim(x-> -7+, sqrt (x+7)) 7. lim(x->infinity, x - sqrt x) 8. lim(x-> infinity, x^3+x^2 - x - 1) 9. lim(t-> 0, sqrt (1/t^2)) 10. lim(x-> 0, 10/x^(1/5)) 11. lim(y-> 0+, y^(- 1/5)) 12. lim(x-> 0+, 3/(x^2 - 3x)) 13. lim(x-> 0+, (x^2 - 1) / x) 14. lim(x-> 5, 1 / (x - 5)^3) 15. lim(x-> 0- , (x^2 - 1) / x) 16. lim(x->infinity, x - x^3) 17. lim(x->-infinity, root(3, x^2)) 18. lim(x-> infinity, sqrt(x^3+x+1) / (x^2+sqrt(x)+1)) 19. lim(x-> 0, (1/x+x^2) / (x^2+1)) 20. lim(x-> 1+ , (x^3 - 1) / (x - 1)^3) 21. lim(x-> 2, (x^3 - 8)^2 / (x^2 - 4)^3) 22. lim(x->-3+, x^2/(x^2 + 5x+6)) 23. lim(x-> - 1- , (x^4 - 1) / (x+1)^2) 24. lim(x-> 1, (x^2 - x+1) / (5x - 5)) 25. lim(x-> 1, (x^2+x+1) / (x^2 - 1)) 26. lim(x->0, (x+1)/x^2 - (x-1)/x^2) 27. lim(x-> infinity, x / (x+1)+x^3 / (x^2+1)) 28. lim(x-> 0+, sqrt x + 1/ sqrt x) 29. lim(x-> 2+ , 1 / sqrt (x^2 - 4)) 30. lim(x-> 2-, x / sqrt (4 - x^2)) 31. lim(x-> infinity, sqrt x + 1 / sqrt x) 32. lim(x-> 5- , sqrt (x / (5 - x))) 33. lim(x-> 3- , sqrt (9 - x^2) / (x - 3)) 34. lim(x-> infinity, root(3, (9+x^3) / (x(x+1)))) 35. lim(x-> infinity, sqrt (9+x^2) - root (4, x)) 36. lim(x-> 1+, 1/(sqrt (x^2 - x) + x)) 37. lim(x-> 0- , 1/ abs x) 38. lim(x-> - 2+ , abs (x / (2+x))) 39. lim(x-> 1- , (x+1) / abs (x - 1)) 40. lim(t->(pi/2)+, sec t) 41. lim(x-> pi/2 , sec^2 x) 42. lim(x-> pi/2- , csc (2x)) 43. lim(x-> pi/2- , 10 / cos x) 44. lim(x-> 0, (cos x) / x) 45. lim(theta-> pi, theta / sin^2 theta) 46. lim(theta->pi/2-, (tan theta) / theta) 47. lim(x-> pi/2, (1 - cos x) / cos x) 48. lim(x-> 1, (sin x) / (x (x^2 - 1))) 49. lim(x-> infinity, x cos (1 / x^2)) 50. lim(x-> 0, x sin (1/x^2)) 51. lim(t-> 1- , ln (1 - abs t)) 52. lim(x-> 1+ , ln (ln x )) 53. lim(x->infinity, (sqrt x + 3)/(x+3)) 54. lim(x->infinity, x/sqrt(x-1)) 55. lim(x->infinity, (sqrt x)/ root(3, x)) 56. lim(x->0, ((x+1)/x^2)^2 - ((x-1)/x^2)^2) @Each term goes to infinity, so you have to combine them before evaluating the limit. 57. lim(x-> - 1/4+ , (x - 1) / (4x+1)) 58. lim(x-> 1/2+ , (4x - 3) / (x - 1/2)) 59. lim(t-> infinity, root (3,t) sin (1/t)) 1. sum(1,i,1,7) 2. sum((-1+k/2),k,1,6) 3. sum((3 + j),j,1,3) 4. sum(k(k-1),k,1,10) 5. sum((i-3)(i+2),i,2,7) 6. sum(2(n-1),n,1,3) 7. sum((1-m/10),m,1,10) 8. sum((i/14)^2,i,1,7) 9. sum(k/2(3 + k/2),k,1,8) 10. sum((i^2-5),i,-3,3) 11. sum((j^2+2j),j,1,20) 12. sum((m^2-3m+1),m,-2,1) 13. sum(i(i-2),i,3,6) 14. sum(1/5(n/5)^3,n,1,5) 15. sum(1/5(k/5)^2,k,1,5) 16. sum(k^3,k,1,5) 17. sum(k^4,k,1,10) 18. sum((i+i^2-i^3),i,1,8) 19. sum((3i^3+i),i,1,10) 20. sum(i(i-1)(i-2),i,1,8) 21. sum(1/(n+1),n,1,5) 22. sum(1/(k-1),k,2,8) 23. sum(1/k(k+1),k,1,10) 24. sum(5i/(i+1),i,1,5) 25. sum(1/i^2,i,1,8) 26. sum((i-1)/(i+1),i,1,4) 27. sum((i+1)(i-1)/i,i,1,6) 28. sum(2^i , i,0,4) 29. sum(sqrt i,i,1,5) 30. sum( 1 +(-1)^i,i,1,10) 31. sum(i/6(-1)^i,i,1,6) 32. sum((-1)^i/(i-1),i,2,6) 33. sum((-1)^i/(i+1),i,1,5) 34. sum(cos(i pi/6),i,1,6) 35. sum(sin(i 2pi/6),i,0,5) 36. sum((-1)^i cos(i pi/2),i,0,5) 37. sum( cos(k pi), k,0,8) 38. sum(3^i,i,0,4) 39. sum(3^(-i),i,1,5) 40. sum(2^i/(i+1),i,0,10) 41. sum(1-5i,i,1,n) 42. sum(k/n,k,1,n) 43. sum(x^k,k,5,8) 44. sum(j/n^2,j,1,n) 45. sum(k^2/n^3,k,1,n) 46. sum(3i^2+i,i,1,n) 47. sum((3+2k)^2,k,1,n) 48. sum(i^3+2i^2+i+1,i,1,n) 49. sum((2+i/n)(2/n),i,1,n) 50. sum((-1)^n,n,1,20) 1. integral (7, x) 2. integral (7, x, -1, 1) 3. integral (3x+4, x) 4. integral ( 3x - 1,x,0, 2) 5. integral (t/2 - 2, t, -2, 2) 6. integral (x^2, x, -1, 1) 7. integral( x^2,x,0,t) 8. integral (x^3, x, -1, 1) 9. integral( x^2,x,0, sqrt(t)) @The square root term is a constant. 10. integral (x^2+1, x, 0, 1) 11. integral( (x^2 + 2x + 1),x) 12. integral (3x^2+4, x, -2, 2) 13. integral (9t^2 - 3t+4, t) 14. integral (x^2+2x+1, x, 1, 3) 15. integral( x^2+x+1,x,1,4) 16. integral( 3x^2-6x+5,x) 17. integral (x^2+2x+1, x, -3, -1) 18. integral( x^2+4x+4,x,0, 9) 19. integral( (x^2 + 2x + 1),x) @Don't factor the integrand. 20. integral( 5x^3-3x^2-x+5,x) 21. integral (6 - x - x^3, x) 22. integral (x^3 - x^2 + x - 7, x, -1, 2) 23. integral (1 - x - x^2 - 3x^3, x) 24. integral (a x^2 - x^3, x, 0, a) 25. integral (4x^3 - 6x^2 - 4x, x, -1, 2) 26. integral (2t^3 - t^2 - 3t+6, t, -2, 0) 27. integral (3t^3 - 2t - 1, t) 28. integral (u^3 - u, u, -x, x) 29. integral (3x^4 - 6x+6, x) 30. integral (3v^5 - 5v^5, v) 31. integral (x^4+2x^2+1, x, -1, 2) 32. integral (x^7 - 3x^4 + 3, x) 33. integral( x^33 -2x^11 -1,x) 34. integral ((7x - 1)^2, x) 35. integral ((3x+1)^2, x) 36. integral ((x+1)^4, x) @Expand the integrand. 37. integral ((x+1)^4, x, -2, 0) @Expand the integrand. 38. integral ((2x + 1)^3, x) 39. integral ((x+1)(x - 3), x, -2, 2) @Multiply out the integrand. 40. integral (x(x^2+8), x) @Multiply out the integrand. 41. integral ((x - 1)(4x^2 - 3), x) 42. integral (3x^2(x^3+1), x) 43. integral (2x(x^2+1), x, 1, 4) @Multiply out the integrand. 44. integral ((x^2 + 3)(x^2 - 3), x) 45. integral ((x^2+2x+1)^2, x, -3, -1) 46. integral( (x^2-1)/(x-1),x) @Factor the numerator and cancel. 47. integral( (x^3-1)/(x-1),x) @Factor the numerator and cancel. 48. integral ((x^4 - 1) / (x^2+1), x) @This simplifies to a polynomial, 49. integral ((x^6 - 1) / (x^2 - 1), x) @This simplifies to a polynomial. 50. integral ((x^6+1) / (x^2+1), x) @This simplifies to a polynomial. 1. integral (x^(3/5), x, 1, 32) 2. integral (1 - 2x+3x^2, x, 0, 2) 3. integral ( 5y^4 -6y^2 + 14,y,-3,3) 4. integral ( u^2 - u, u, -x, x) 5. integral (1 / x^5, x, -3, -1) 6. integral (1/y^3+10/y^4+1/y^5, y) 7. integral ((x^3+1) / x^2, x) 8. integral( 1/t^2 + 1/t^4,t,1, 2) 9. integral (1 / sqrt x, x ,1, 2) 10. integral ((t - 4) sqrt t, t) 11. integral ((x+1/x)^2, x) 12. integral( root(4, x^5) + root(5, x^4),x,0,1) 13. integral ((x - x^5) / sqrt x, x) @Simplify the integrand first. 14. integral (x^(-2/3)+4x^(-1/3), x) 15. integral ((7x - 7) / root (3, x), x) 16. integral (root (3, x) - 1 / root (3, x), x, 1, 8) 17. integral ((x- 1) / x^2, x) @Break the integrand apart into two fractions. 18. integral( (t^6 - t^2)/t^4,t,1,2) @Simplify the integrand first. 19. integral( (x^2+1)/sqrt x,x,1,2) @Simplify the integrand first. 20. integral( (x^3-1)^2, x,0, 2) @Expand the integrand. 21. integral( (x-1)(3x+2),x,1,-1) @Multiply the integrand out. 22. integral( (x-2)(x+1),x) @Multiply the integrand out. 23. integral( (2x-1)(x+3),x) @Multiply the integrand out. 24. integral (1 / (3x)^3, x) 25. integral (y^2 root (3, y), y) 26. integral( (1+ sqrt x)^2,x) 27. integral( (x - 1)^3,x,-1,1) 28. integral ((x+2)^4, x) 29. integral ((x^2 - 4) / (x+2), x) 30. integral ((t^2 - 1) / (t+1), t) 31. integral ((t^3 - 1) / (t - 1), t) 32. integral ((t^4 - 1) / (t^2+1), t) 33. integral (cos x, x) 34. integral (sin x, x) 35. integral( 1+cos t,t, 0,x) 36. integral (x - cos x+1, x) 37. integral (sin x cos x, x, -pi, pi) 38. integral (1 / sin^2 x, x) 39. integral (csc^2 theta, theta) 40. integral ((sec x) / cos x, x) 41. integral (tan^2 theta+1, theta) 42. integral (csc x cot x, x) 43. integral( sec x tan x, x,0, pi/3) 44. integral (sec^2 x - sec x tan x, x) 45. integral ((sec^3 x+tan x) / sec x, x) @Simplify the integrand first. 46. integral (abs x, x, -1, 1) @Many textbooks would break up the interval of integration to get rid of the absolute value, but this is actually not necessary, because $abs(x)$ can be integrated directly. 47. integral (x abs x, x, -1, 1) 48. integral (1+abs x, x, -1, 1) 49. integral (abs (x - 1) abs (x+1), x, -2, 2) @Break up the interval of integration. 50. integral( abs(x-x^2),x,-1,2) @Break up the interval of integration. 51. integral( u(sqrt u + root(3, u)),u,0,1) @Multiply the integrand out. 52. integral( abs(x^2-1),x,-2,3) @Break up the interval of integration. 53. integral( abs(x-x^2),x,-1,2) @Break up the interval of integration 54. integral (abs (sin x), x, -pi/2, 0) @Break up the interval of integration. 1. diff (integral(2,t,-2,x),x) 2. diff (integral(sqrt (1 - t),t, 0,x),x) 3. diff (integral(t^(11/5),t, x,0),x) 4. diff (integral(sqrt(t^5+1),t,1,x),x) 5. diff (integral((1 - sqrt t) / sqrt t, t,1,x),x) 6. diff (integral((2 - sqrt(t))^9,t,x,0),x) 7. diff (integral(t / sqrt (1 - t^3), t,-3,x),x) 8. diff (integral(3t^10 (t^3+1)^11,t, x,2),x) 9. diff (integral(e^(t^2), t, 0,x),x) 10. diff (integral(t ln t,t, e,x),x) 11. diff (integral(1 - cos t, t, 3, x),x) 12. diff (integral(sin (t^3),t, 0,x),x) 13. diff (integral(sin(t^7),t,0,x),x) 14. diff (integral(1 - t^3, t, x,cos x),x) 15. diff (integral(abs (t^2 - 4),t, 4,x),x) 16. diff (integral(csc t tan^3 t,t,1,-x),x) 17. diff (integral(root (3,1+t+t^2), t,2,1+x),x) 18. diff (integral((1+t^2)^(1/5), t,1,5x),x) 19. diff (integral(10,t,-2,x^3),x) 20. diff (integral(t^2+t+1, t,-1,x^2),x) 21. diff (integral(t^2/(t^2+1),t,1,sqrt x),x) 22. diff (integral(sqrt (1+t^2),t,0,x^3),x) 23. diff (integral((t^2+3)^18, t,10,1+x+x^2),x) 24. diff (integral((1+t^2)^3/4, t,1,x^6),x) 25. diff (integral(sqrt (t+sqrt t), t, a,x^4),x) 26. diff (integral(csc t, t, pi/2,x^3),x) 27. diff(integral(x^99,x),x) 28. diff (integral((sin t)/t,t,x^3, 2pi),x) 29. diff (integral(sin^3 (t^2), t, 0,sqrt x),x) 30. diff (integral(abs (t+sqrt t), t, 4,4x^2),x) 31. diff (integral((2t - 1) / (t^2+2t+1), t,1,x sin x),x) 32. diff (integral((t^10 - 1) / (t^2+1),t,1,cos x),x) 33. diff (integral(t^2+t+1, t,1,sec x),x) 34. diff (integral(t cos t, t, pi,tan x),x) 35. diff (integral(t cos t^2, t, 5, tan x),x) 36. diff (integral((1 - t^2)^2, t,1+x,1+x^2),x) 37. diff (integral(1/t, t,3x,5x),x) 38. diff (integral(1/t, t,sin x,x^2),x) 39. diff (integral(1 /(1+t^2), t,-x,3x^2+1),x) 40. diff (integral((t^2+1)^7, t,x,x^2),x) 41. diff (integral(sqrt (t^4+2), t, -x,x),x) 42. diff (integral(1 / sqrt (1 - t^2), t, x ,x^2),x) 43. diff (integral(tan t, t,sin x,x),x) 44. diff (integral(sin t, t, -x, 1-x^2),x) 45. diff (integral(tan^3 t, t,-x,-x^2),x) 46. diff (integral(t^2 cos (t^3), t,2x,4x),x) 47. diff (integral(sin(t^2),t,root(5,x), -x),x) 48. diff (integral(1 - t^2, t,cos x,sin x),x) 49. diff (integral(1 /(1 - t^5), t,-x^2,x^2),x) 50. diff (integral(sqrt (1 - t^2),t, 0,1),x) 51. diff(integral(e^x^2 sin x,x),x) 1. integral (3t/(t^2+1)^2, t ) 2. integral( 1/(3x-1)^2,x) 3. integral ( (1 - y)^11, y ) 4. integral (x^3 (1+x^4)^3, x ,0,1 ) 5. integral ((6 -x)^(-3), x,1,2 ) 6. integral ((2x+1)^-1/3, x) 7. integral (t (3 - t)^(1/3), t, 1, 3 ) 8. integral (3t (1-3t^2)^10, t ) 9. integral (y (y+1)^15, y) 10. integral ( (y^3+6y^2 - 12y+5) (y^2+4y - 4), y, 0, 1 ) 11. integral ( t^9 sin (t^10), t) 12. integral (x sin(x^2), x, 0, sqrt pi) 13. integral ((x^2) /(x^3 +1), x, 1,3) 14. integral ( (x+3)/sqrt (x+1), x, 0, 1 ) 15. integral (x^2 sqrt (x^3+1), x) 16. integral (x (x^2+1)^(1/3),x, - sqrt 7, 0) 17. integral( x^3(1+x^4)^2,x) 18. integral( x(3x^2+1)^(1/2),x) 19. integral( 1/(1-t)^8,t) 20. integral(x^2 root(3, x^3+1),x) 21. integral( sqrt(1+2x),x,0,4) 22. integral(2 x/sqrt(1+2x),x,0,4) 23. integral (1/(sqrt x (1+ sqrt x)^2), x) 24. integral ( sqrt (1+ sqrt x)/ sqrt x , x) 25. integral ( 1/ (x + sqrt x), x ) 26. integral (x^(1/3) cos x^(4/3) ,x) 27. integral ( x^2 sqrt (x+3) , x ) 28. integral ( x sqrt (x-1), x ) 29. integral ((x+2)/ (x^2+4x+3)^3, x ) 30. integral ((2 - x^2)/(x^3 - 6x + 1)^5, x ) 31. integral (sqrt (10 - sqrt x), x, 1, 9 ) 32. integral ( y^(1/2) (1+y^(3/2))^7, y ) 33. integral ((1+sqrt x)^4 /sqrt x, x, 1, 4 ) 34. integral (x^(n-1) sqrt (a+bx^n) , x ) 35. integral ( (x^2+x) sqrt (2x^3+3x^2+2), x) 36. integral( sin 6t,t,0, pi/2) 37. integral (sin (3 - 5x), x ) 38. integral (sec (2x) tan (2x), x ) 39. integral ( sin(sqrt theta)/sqrt(theta), theta) 40. integral ( tan theta sec^3 theta, theta) 41. integral (cos (3x)/ sin^3 (3x), x ) 42. integral (csc^2 (2 theta) cot (2 theta), theta) 43. integral (sec^3 theta tan theta, theta ) 44. integral (sin theta cos theta, theta ) 45. integral ( x cos (x^2), x) 46. integral ((sin (1/x))/ x^2, x) 47. integral (cos^4 t sin t, t ) 48. integral( sec^2 7 theta,theta) 49. integral( (tan sqrt x) / sqrt x,x) 50. integral( tan^2 x sec^2 x,x ) 51. integral (sqrt (tan x) sec^2 x, x, 0, pi/4 ) 52. integral ( (cos x)/ sqrt (2+ sin x), x, 0, 2pi) 53. integral (sin (sqrt y)/ sqrt y, y ) 54. integral (sin^2 x cos x, x ) 55. integral ( x^(-3) cos (x^(-2)), x ) 56. integral ((cos x) sqrt (sin x ), x, 0, pi/2) 57. integral ((sin (sqrt t) cos (sqrt t))/ sqrt t, t ) 58. integral ((sin x)/(1+cos x)^4, x, 0, 1) 59. integral ( 3 e^x/ (1+3 e^x), x) 60. integral (e^(2x) sin (1+ e^(2x)), x ) 61. integral( sin x root(3, cos x),x) 1. integral( x cos x, x,0, pi) 2. integral (x cos (pi x), x, 0,1/2) 3. integral( x sin 3x,x) 4. integral (theta^2 sin (2 theta), theta, 0,pi/2) 5. integral (x^2 cos x, x,0,pi) 6. integral (x^3 sin x, x) 7. integral (x^3 cos x, x) 8. integral (sin (sqrt x) ,x) 9. integral (cos (sqrt x), x) 10. integral (x sec^2 x, x) 11. integral( x csc^2 x,x) 12. integral( x sec x tan x,x) 13. integral( x sec^2 x,x) 14. integral (x^3 cos (x^2), x) 15. integral( (x+1) e^x,x) 16. integral (x e^(-2x), x) 17. integral (x e^(1-5x), x) 18. integral (x^4 e^x, x) 19. integral ((1+3x+x^2) e^x, x) 20. integral (x^2 e^(2x), x) 21. integral( x^3 e^(x^2),x) 22. integral (x^5 e^x, x) 23. integral (x^3 3^x, x) 24. integral (t^2 4^t, t) 25. integral (e^(sqrt x), x) 26. integral (e^(root(3, x)), x) 27. integral (e^(sqrt (3s+9)), s) 28. integral (ln (1+4x), x) 29. integral (ln (x)/x^2, x) 30. integral (x^(3/2) ln x, x, 1,4) 31. integral (x^3 ln x, x) 32. integral (x^2 ln (x^3), x) 33. integral( t^2 ln t,t) 34. integral (ln (1+x^2)/x^2, x) 35. integral ((ln (x))^3, x) 36. integral ((ln (x))/(x+9)^2, x) 37. integral (x ln (4+x^2), x, 0,1) 38. integral (x (ln x)^2, x) 39. integral (ln (x+x^2), x) 40. integral ((ln (x+1))/ sqrt(x+1), x) 41. integral (sqrt x ln (x^3), x) 42. integral (sqrt x ln x, x) 43. integral (e^x ln (1+e^x), x) 44. integral (sin (ln x), x) 45. integral (cos (ln x), x) 46. integral( cos(4 ln x),x) 47. integral( cos x ln (sin x),x) 48. integral (e^x sin(x/2), x) 49. integral (e^(2x) (3-sin (4x)), x) 50. integral (e^(2x) cos (pi x), x, 0, 1) 51. integral ((sin x)/e^x, x) 1. integral (x (2x+3)^99, x) 2. integral ((x^3 -1)/(x^2+1), x) 3. integral ((2x +1)/((x^2+1)(x+3)), x) 4. integral (x^2/sqrt (1-x), x) 5. integral (x^3 sqrt (1-x^2), x) 6. integral (x^7/(1+x^4)^(3/2), x) 7. integral( xe^(2x),x) @better done by substitution 8. integral( x^2 e^(-x), x,0,1) @Usually in a product of a polynomial and transcendental, you differentiate the polynomial and integrate the transcendental if you can. 9. integral( t e^(-t),t,0,1) 10. integral( x 5^x,x) 11. integral( (2x+3) e^x,x) 12. integral (x^2e^(-x^3), x) 13. integral( x^3 e^(x^2),x) @First perform a substitution. 14. integral( x^5 e^(x^2),x) 15. integral( e^(sqrt x),x) @First perform a substitution. 16. integral( t^3 e^t,t) 17. integral( x ln x,x) @Differentiate the logarithm, because its derivative is a rational function. 18. integral( (ln x)^2,x) @If any factor is a logarithm or a power of a logarithm, choose that for the part to differentiate. 19. integral( t^2 ln t,t) 20. integral( (sqrt t) ln t, t,1,4) 21. integral( ln(sqrt x),x) 22. integral ((ln x)/x, x) 23. integral (1/(x ln x), x) 24. integral( x cos x,x) 25. integral( x sin 4x,x) 26. integral( x cos 2x, x,0, pi/2) 27. integral( sin(3x) cos(5x),x) @Better done by using a trig identity first. 28. integral( sin(2x) sin(4x),x) @Better done by using a trig identity first. 29. integral (sin^2 theta, theta) 30. integral (cos (x) sin^2 (x), x) 31. integral (sin^5 (2u), u) 32. integral (sec^5 x, x) 33. integral( theta sin theta cos theta,theta) 34. integral( theta sec^2 theta,theta) 35. integral( x csc^2 x, x,pi/4, pi/2) 36. integral( x^2 cos 3x,x) 37. integral( x^2 sin 2x,x) 38. integral (x sin (x^2), x) 39. integral( x^5 cos(x^3),x) 40. integral( sin(sqrt x),x) 41. integral( arccos x, x,0, 1/2) 42. integral( arcsin x,x) 43. integral( arctan x, x) @The derivative of arctan is a rational function. 44. integral( x arctan x,x) @Usually when one factor is a polynomial and the other is transcendental, we differentiate the polynomial. But the derivative of arctan is a rational function, so if you differentiate the arctan, the transformed integrand will be a rational function. 45. integral( e^(2 theta) sin(3 theta),theta) 46. integral( e^(-theta) cos(3 theta),theta) 47. integral( cos(ln x)/x,x) 48. integral( cos x ln(sin x),x) 49. integral( sin(ln x),x) @Note this is not a product of sin and ln, but a composition of functions. First perform a substitution. 50. integral( cos(ln x),x) 51. integral(abs(sin x),x,0,3pi/2) 52. integral(abs(sin x),x,0,5pi/2) 1. lim(x->infinity, 2^x) 2. lim(x->infinity,pi^(-x)) 3. lim(x-> - infinity, (sqrt 2)^x) 4. lim(x->infinity,5^(2x+5)) 5. lim(x-> infinity, 2 ^(- 8 x)) 6. lim(x-> - infinity, 3^(- x^2)) 7. lim(x-> - infinity, e^(5x^3)) 8. lim(x-> 1/2, e^(5x^3)) 9. lim(x-> infinity, e^( sqrt x)) 10. lim(x-> 0+, e^( -1/sqrt x)) 11. lim(x-> 0+, e^(- 1/x)) 12. lim(x-> infinity, 1 / (e^x+1)) 13. lim(x-> 0+, e^(-1/x)+1/e^x) 14. lim(x-> infinity, e^(1 / x)+1 / e^x) 15. lim(x-> - infinity, e^(1/x)+1/e^x) 16. lim(x-> - infinity, 1/x - e^(- x^4)) 17. lim(x-> infinity, sqrt (e^(-x))) 18. lim(x-> 0, sqrt (1+e^(2x))) 19. lim(x-> pi -, e^( cot x)) 20. lim(x-> - infinity, sqrt (1+e^(2x))) 21. lim(x-> infinity, sqrt (1+ e^(2x))) 22. lim(x-> infinity, root(4, 16+ e^(-x))) 23. lim(x-> infinity, e^(e^(-x))) 24. lim(x-> infinity, ln (1/x +(e^(1/x)))) 25. lim(x-> pi-, e^(cot^2 x)) 26. lim(x-> pi+, e^(cot x)) 27. lim(x-> pi, e^( sin 3x)) 28. lim(x-> pi+, 2^(sin x)) 29. lim(x-> pi/2-, e^(tan x)) 30. lim(x-> 0, (pi)^(sin x)) 31. lim(x-> 0, cos (pi/4e^(-x))) 32. lim(x-> infinity, sin^2 (e^(-x))) 33. lim(n-> infinity,(1+1/n)^n) 34. lim(n-> infinity,(1+1/(n+1))^n) 35. lim(n->-infinity,(1+1/n)^n) 36. lim (x-> infinity, (1 - 1/x)^x) 37. lim (x-> infinity, (1+3/x)^(-x)) 38. lim (x-> infinity, (1+1 / (2x))^x) 39. lim (x-> infinity, (1-2/x)^(-x+3)) 40. lim (x-> infinity, (1+1/x^2)^x) 41. lim (x-> infinity, (1 - 1/x^2)^(x^2-1)) 42. lim (x-> infinity, (1+1 /(x^2-2x))^x) 43. lim (x-> - infinity, (1+1/x)^(x^2)) 44. lim (x-> infinity, (1+1 / (pi x))^(3x)) 45. lim (x-> - infinity, (1+1/x)^(x^2)) 46. lim (x-> infinity, (1+1 / x^10)^x) 47. lim (x-> infinity, (1+1 / (sqrt(2 x)))^(sqrt(2x))) 48. lim(h->-infinity, e^(-h) ln(e^h+1)) 49. lim (x-> 0-, (1+ sin x)^(csc x)) 1. lim(x->0, sin(x)/x) 2. lim(x->0, tan(x)/x) 3. lim(x->0, (1-cos x)/x) 4. lim(x->0, (1-cos x)/x^2) 5. lim(x->0, (e^x-1)/x) 6. lim(x->0, ln(1+x)/x) 7. lim(x->0, (sin 5x)/(sin 3x)) 8. lim(x->0, (sin^2 x)/x) 9. lim(x-> infinity, (x^2) / ln x) 10. lim(x->1, (sin (x pi)) / (x - 1)) 11. lim(x->0, (1 - cos x) / x^2) 12. lim(x->0,(e^(2x) - 1)/(sin x)) 13. lim(x->0, (3^x - 1) / (5^x - 1)) 14. lim(x->0, (ln^2 (1+x)) / x) 15. lim(x->infinity, (3^x) / 5^x) 16. lim(x->0, (e^(4x) - 1) / (2x)) 17. lim(x->0+, (x^(-1/2)) / ln x) 18. lim(x->0+,(ln x)/sqrt x) 19. lim(x->infinity, (ln (ln x)) / ln x) 20. lim(x->0, (ln (cos x)) / sin x) 21. lim(x-> e-, (1 - ln x) / (x - e)^2) 22. lim(x->infinity, (ln (x^2+10)) / ln x) 23. lim(x->1-, (ln x) / arccos x) 24. lim(x->2,(root(3, x) - root(3, 2))/(x-2)) 25. lim(x->3, (sqrt (x+1) - 2) / (x^2 - 9)) 26. lim(x->0,(e^x-1)/sin x) 27. lim(x->0, (1 - e^(3x)) / sin(3x)) 28. lim(x->infinity, (x^10) / e^x) 29. lim(x->2, ( sin (4 - 2x)) / (x^3 - 8)) 30. lim(x->infinity, (x^(3/2)+5x - 4) / (x ln x)) 31. lim(x->infinity,(ln ln x)/sqrt x) 32. lim(x->0, x/(arcsin 5x)) 33. lim(x->0, (arctan x - x) / x^2) 34. lim(x->0, (sinh (x^2) - tanh (x^2)) / x^3) 35. lim(x-> pi/2,(cos x)/(x - pi/2)) 36. lim(x->0, ( sqrt (1+x) - sqrt (1 - x)) / x) 37. lim(x->infinity, 6 (x - sqrt (x^2+x))) 38. lim(x->- infinity, x+sqrt (x^2+1)) 39. lim(x->1,1/(ln x) - 1/(x-1)) 40. lim(x->pi/2+, ((pi/2) - x) tan x) 41. lim(x->0+, (sin x) ln (sin x)) 42. lim(x->-infinity,xe^x) 43. lim(x->infinity, x e^(-x)) 44. lim(x->0+, x^3 ln x) 45. lim(x->infinity,x^3e^(-x^3)) 46. lim(x->1+, (ln x) tan (pi x/2)) 47. lim(x->infinity, x^(1/x)) 48. lim(x->0+, x^(1/ln x)) 49. lim(x->1/2-, (1 - 2x)^ln (2x)) 50. lim(x->0+, x^(sin (x^2))) 51. lim(x->infinity, (1+x^(-2))^x) 52. lim(x->infinity, (x^3+1)^(1/ln x)) 53. lim(x->infinity,(1+3/x)^(5x)) 54. lim(x->infinity,((2x-1)/(2x+1))^(x+1)) 55. lim(x->infinity,(x/(x - 1))^x) 56. lim(x->0+,x^(2sin x)) 57. lim(x->0+,(-ln x)^x) 58. lim(x->infinity, (e^x+1)^(1/x)) 59. lim(x->pi/2, abs (sec x)^(cos x)) 60. lim(x->infinity, (cosh x)^(1/x^2)) 61. lim(x->infinity,x(ln(x+5)-ln x)) 62. lim(x->0+, (e^(-1/x)) / x) 63. lim(x->0+, (sin x) e^ (1/x)) 64. lim(x->0+, (ln (1/x))^(x^2)) 65. lim(x->(pi/2)-,(tan x)^( cos x)) 66. lim(x->0+, (cot x)^sin x) 67. lim(x-> infinity, (e^(2x) - x^2) / (e^(2x) + x^2)) 68. lim(t->2, sin (t^2 - 4) / (t - 2)) 69. lim(t-> 0, sin^2 (t/2) / sin (2t)) 1. lim(x->infinity, (3x+1)/(2x-6)) 2. lim(x->infinity, (x^2-2x+3)/(3x^2+x)) 3. lim(x->infinity, (x^3+x^2-x-3)/(x^3+x+1)) 4. lim(x->infinity, (3-x)/(x^2-2x-8)) 5. lim(t->-infinity,(6t^2 + 5t)/((1-t)(2t-3))) 6. lim(x-> infinity, x / (x^2 - 1)) 7. lim(x-> infinity, (5x + 10) / (x (x+3))) 8. lim(x-> infinity, (x^2 - x) / (x^3-x^2)) 9. lim(x-> infinity, x^3/(x + 5) - x^2) 10. lim(x->infinity, (x^3+x+1)/(x^4+x^2+1)) 11. lim(x->infinity, (x^4+x^2+1)/(x^5-x^3+x)) 12. lim(x->infinity, (x^7-1)/(x^6+1)) 13. lim(x->infinity, (5x^3+4x)/(2x^3-x^2+1)) 14. lim(x->infinity, (11-x^5)/(x^2+x+1)) 15. lim(x->infinity, (5x^3+5x+1)/((1-x)(2x-3))) 16. lim(x->infinity, ((1-x)(2+x))/((1+3x)(2-3x))) 17. lim(x-> infinity, (9x^3 - x) / (x^3 +1)) 18. lim(x-> infinity, (x-2) / (x-3) + (x^3+2) / (x^3-3)) 19. lim(x-> infinity, (x^2+1) / x^4) 20. lim(x-> infinity, x^2/(x^3+1) + x^2/(x^3 - 1)) 21. lim(t-> infinity, t / (t^(1/2) - 2t^(-1/2))) 22. lim(x-> infinity, (x + sqrt x) /(x- sqrt x)) 23. lim(x-> -infinity, sqrt(3x^4-x)/(x^2-10)) 24. lim(x-> infinity, sqrt(x^2+4x)/(4x+1)) 25. lim(x-> infinity, sqrt(x^2 - 1)/sqrt(x^2+1)) 26. lim(x-> infinity,sqrt(x^2+5x)/(1- x)) 27. lim(x->infinity,sqrt((2x^2-1)/(x+8x^2))) 28. lim(x->infinity, sqrt(1+4x^2)/(4+x)) 29. lim(x->-infinity,sqrt(x^2+4x)/(4x+1)) 30. lim(x->-infinity, sqrt(x^2+x+1) + x) 31. lim(x-> infinity, 8x/ sqrt (4x^2-1)) 32. lim(x-> infinity, (sqrt x) / (4 sqrt x + x)) 33. lim(x-> infinity,sqrt(1+x)/sqrt(x-1)) 34. lim(x-> infinity, sqrt(x^2+x -1) /(x-1)) 35. lim(x-> infinity, sqrt (x^2/(1+3x^2))) 36. lim(x-> infinity, sqrt(4 + x^2)/(1+4x)) 37. lim(x-> infinity, sqrt (1+x^2) / (x-1)) 38. lim(x-> infinity, 2x / sqrt (4x^2 - 1)) 39. lim(x-> infinity,sqrt((x^2-1)/(x+7x^2))) 40. lim(x-> -infinity, x / sqrt (4x^2+5)) 41. lim(x-> -infinity, sqrt ((1 - x) / (3 - x))) 42. lim(x-> -infinity,root(3, x+1) / root(3, x - 1)) 43. lim(x->infinity,root(3, 1+x)/root(3,x)) 44. lim(x-> infinity, root(3, x^2+6)/(x+1)) 45. lim(x-> -infinity, root(3, x^3+8)/(x - 1)) 46. lim(x-> -infinity, (8+x^2) / (x (x - 1))^(1/3)) 47. lim(x-> infinity, (x^2+x+1)^(5/3) / (x-1)^4) 48. lim(x-> infinity, x sqrt (9x^2 +1) / sqrt (3x^3 +1)) 49. lim(x-> infinity, (16 - x^2)^(1/3) / (4 - x)) 50. lim(x-> infinity, (2x^(2/3)-x^(1/3)+1)/(x^(1/3)-1)^2) 51. lim(x-> infinity, x/sqrt(x^2+1) + 2x/sqrt(x^2-1)) 52. lim(x-> infinity, root(3, x^4+1)/(x+1) + root(3, x^4-1)/(x-1)) 1. lim(x->-1, (1+3x - x^2) / (x^2+x+3)) 2. lim(x->0, (x^2 - 5x) / (x^2 - 2x)) 3. lim(x->infinity, (x^2(1 - 5x)) / (5x^3+x+1)) 4. lim(x->-3-, (3 abs x +9) / (x^2 - 9)) 5. lim(x->infinity, x^2 / (x+2) - x^3 / (x^2+2)) 6. lim(x->- infinity, x^2 / (1+x)+x^2 / (1 - x)) 7. lim(x->infinity, ((3x^2+1) / (10+3x^2))^x^2) 8. lim(x->0-, sqrt (9+ sqrt(-x))) 9. lim(x->5/2-, x^2 - sqrt(5 - 2x)) 10. lim(x->3-, 1 / (5 - sqrt (x^2+16))) 11. lim(h->0, (sqrt (x+h) - sqrt x) / h) 12. lim(x->infinity, sqrt x - sqrt(x^2 - x - 1)) 13. lim(x->infinity, (x - 10) / sqrt(x^2+100)) 14. lim(t->81, (t^(3/4) - 27) / (t^(1/2) - 9)) 15. lim(x->-1, (4 - root(3, (x^2 - 6x+1)^2)) / (2 - root(3, (x^2 - 6x+1)))) 16. lim(x->0+, (1-x)^(4/x)) 17. lim(x->0+, (1+sqrt x)^(1/x)) 18. lim(x->-infinity,arctan(x^3)) 19. lim(x->0-, (sin x) / abs x) 20. lim(x->infinity, sqrt x sin (1/x)) 21. lim(x->0, (cos x - 1) / (2x)) 22. lim(x->pi, (1+cos x ) / (x - pi)^2) 23. lim(x->0, (1 - cos 3x) / (5x^2)) 24. lim(x->0,(sin x^2) / (x^4 - 3x^2)) 25. lim(x->infinity, (sin 3x+x) / (2x+3)) 26. lim(x->1, (cos (pi x / 2)) / (x - 1)) 27. lim(x->0, (x^2 - 5x) / sin (3x)) 28. lim(x->0, (cos^2 x - sin^2 x) / x) 29. lim(x->1+, (sin (pi x)) / abs (x - 1)) 30. lim(x->0, 1/x^2 - 1 / (1 - cos x)) 31. lim(x->0+, arctan (1 / x)) 32. lim(x->0, (arcsin (5x)) / (11 x)) 33. lim(x->pi-, arcsin (cos x)) 34. lim(x->0+,(sin x)(ln x)^3) 35. lim(x->0+,ln(tan x)) 36. lim(x->0,csc x ln (1 - x)) 37. lim(x->0+, (e^(1/x) - 1) tan x) 38. lim(t->0, (1 - e^(sin t)) / (1 - e^t)) 39. lim(x->- infinity, (1 - e^x) / (1+e^x)) 40. lim(x->infinity, x e^(-(x^(1/10)))) 41. lim(x->infinity, ln(ln x)/ln x) 42. lim(x->0-, 5^(ln (abs x))) 43. lim(x->0+, e^(root (3, ln (sqrt x)))) 44. lim(t->0, (sinh (5t)) / t) 45. lim(x->0, (sin^5x)/sin(x^5)) 46. lim(x->0+, ln (1 - ln (2x))) 47. lim(x->1+, (x^2 - 1) ln (x - 1)) 48. lim(x->infinity, (1+5/x)^(x/10)) 49. lim(x->0+, (e^x) / ln x) 50. lim(x->0+,(1 - e^(-x)) x^(-1/2)) 51. lim(x->1+, (ln x)^ln x) 52. lim(x->-infinity,e^(x/2) cos x) 53. lim(x->1+,(ln x)^sin x) @The two-sided limit is undefined, because the function isn't defined when $ln x$ is negative and $sin x$ is irrational.54. lim(x->0+,(root(3, x)-1)/(root(4, x)-1)) 55. lim(x->0,(e^(ax)-e^(bx))/x) 1. ln x 2. e^x 3. 2^x 4. a^x 5. sinh x 6. cosh x 7. e^(-x) 8. e^(x+3) 9. x e^x 10. ln (abs x) 11. (1/x) e^x 12. (x^2+3) e^x 13. (3x^3) e^(2x) 1. e^(2x+3) 2. e^(3 - x^3) 3. e^(3 - x - x^3) 4. e^(-1/x) 5. 3 - e^(-(x^2)/2) 6. xe^(x^2) 7. x^2 e^(-x^3) 8. e^(sqrt x+1) 9. e^ sqrt(x+1) 10. e^sqrt x +1 11. e^sqrt x + sqrt (e^x) 12. root(3, x^3+e^(3x)) 13. (e^(x^2)+x)^2 14. (x^2+x+1) e^(2x) 15. 1 / (e^x+e^(-x)) 16. e^(e^x) 17. e^(x+e^x) 18. (e^x - e^(-x))/(e^x+e^(-x)) 19. e^(x - e^x^2) 20. e^(sin (3x)) 21. e^(sin^2 x) 22. e^(sin x +cos x) 23. e^cos (2x) 24. cos (2e^x) 25. cos (e^(2x)) 26. sin (e^(7x)) 27. sin (2e^x) 28. (cos x )e^(-2x) 29. cos (e^(-x^2)) 30. tan (e^x) 31. cos (1 - e^(-x)) 32. sin (x - e^(-x)) 33. x^2 e^(x^2) sin (x^2) 34. e^arcsin x 35. e^arccos (x^2) 36. arctan (e^x) 37. 5^x 38. 4^(-x+x^2) 39. 10^(-sqrt x) 40. 5^ sqrt(1+x) 41. 2^(x^2) 42. x 3^(x^2) 43. 2^x e^(5x) 44. 1 / x 2^ln x 45. 3^(6^x) 46. 2^(cos x) sin x 47. 2^(3^cot x) � 5 . . L ; 5 f . 2 RV 5 � . . �z �V 5 � . 6 � 5 � . : *� 5 . ; k� �V �V � 5 o . 3 R_ 5 � . = �� �_ 5 � . 4 � 5 � . 1 B� 5 . E w �� �� 5 g .
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