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Limits, Continuity and Differentiability

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  1. 51.Let [x] denote the greatest integer function and f(x)=max1+x+[x],2+x,x+2[x],0 x 2.
  2. 52.If x 0lime^ax- (bx)-cxe^-cx21- (2x)=17, then 5a^2+b^2 is equal to
  3. 53.Let f(x)=[x^2-x]+|-x+[x]|, where x R and [t] denotes the greatest integer less than or equ…
  4. 54.If > >0 are the roots of the equation ax^2+bx+1=0, and x (1)/( )lim(1- (x^2+bx+a)2(1- x)^2…
  5. 55.x 0lim(((1- ^2(3x) ^3(4x))( ^3(4x)( _e(2x+1))^5)) is equal to _____________.
  6. 56.Let a_1,a_2,a_3, ,a_n be n positive consecutive terms of an arithmetic progression.
  7. 57.x lim(√3x+1+√3x-1)^6+(√3x+1-√3x-1)^6(x+x^2-1)^6+(x-x^2-1)^6x^3
  8. 58.Let f,g and h be the real valued functions defined on R as f(x)= array|c|c| (x)/(|x|), & x…
  9. 59.Let f(x)=x^3+8; & xThen the number of points, where the function g f is discontinuous, is
  10. 60.Let f(x)= array|c|c| e^x-1 & ,x<0 x^2-5x+6 & ,x 0 array and g(x)=f(|x|)+|f(x)|.
  11. 61.The number of points, at which the function f(x)=max6x,2+3x^2+|x-1| |x^2-(1)/(4)|,x (- , )…
  12. 62.The number of points in the interval [2,4], at which the function f(x)=[x^2-x-(1)/(2)], wh…
  13. 63.If the function f(x)=( ( x)- ( x))/( x- x) is continuous at x=0, then f(0) is equal to ___…
  14. 64.For t>-1, let _t and _t be the roots of the equation ((t+2)^1/7-1)x^2+((t+2)^1/6-1)x+((t+2…
  15. 65.The number of points of discontinuity of the function f(x)=[x^22]-[x],x [0,4], where [ ] d…
  16. 66.Let m and n be the number of points at which the function f(x)=maxx,x^3,x^5, x^21,x R, is …
  17. 67.Answer: 32
  18. 68.Let f(x)=n lim _r=0^n( (x/2^r+1)+ ^3(x/2^r+1)1- ^2(x/2^r+1)) Then x 0lime^x-e^f(x)(x-f(x))…
  19. 69.Let f(x)=array|c|c| 3x, & x2 array where [.] denotes greatest integer function.
  20. 70.Let the function, f(x)=-3ax^2-2, & xbe differentiable for all x R, where a>1,b R.
  21. 71.Let f:(0, ) R be a function given by f(x)=((8)/(7))^( 8x)/( 7x), & 0where a,b Z.
  22. 72.If =x 0^+lim(e^ x-e^x x-x) and =x 0lim(1+ x)^(1)/(2) x are the roots of the quadratic equa…
  23. 73.The value of x 0lim2(1- x 2x√3 3x x^2) is __________.
  24. 74.Let [t] denote the greatest integer less than or equal to t.
  25. 75.Let a>0 be a root of the equation 2x^2+x-2=0.
  26. 76.Let f be a differentiable function in the interval [(0, )] such that f(1)=1 and [ _t x (t^…
  27. 77.If x 1lim(5x+1)^1/3-(x+5)^1/3(2x+3)^1/2-(x+4)^1/2=m√5n(2n)^2/3, where gcd(m,n)=1, then 8m+…
  28. 78.Let x denote the fractional part of x and f(x)= ^-1(1-x^2) ^-1(1-x)x-x^3,x 0.
  29. 79.If x 0limax^2e^x-b _e(1+x)+cxe^-xx^2 x=1, then 16(a^2+b^2+c^2) is equal to ________.
  30. 80.If the function f(x)=(1)/(|x|), & |x| 2 ax^2+2b, & |x|is differentiable on R, then
  31. 81.Let f(x)=r xlim2r^2[(f(r))^2-f(x)f(r)]r^2-x^2-r^3e^(f(r))/(r) be differentiable in (-,0) (…
  32. 82.Let [x] be the greatest integer x.
  33. 83.Let f:(-2,2) R be defined by f(x)= array|c|c| x[x], & -2<x<0 (x-1)[x], & 0 x 2 array where…
  34. 84.Let k and m be positive real numbers such that the function f(x)=3x^2+kx+1, & 00.
  35. 85.Let a Z and [t] be the greatest integer t.
  36. 86.The number of points, at which the function f(x) =, 2x + 1, - 3| x + 2, +, x2 + x - 2 |, x…
  37. 87.Let [x] denote the greatest integer function.
  38. 88.Let f(x)=6x,,2+3x^2+|x-1||x^2-14|, x(-,).
  39. 89.Let f(x)=casesx^3+8, & x<0 x^2-4, & x0cases and g(x)=cases(x-8)^1/3, & x<0 (x+4)^1/2, & x0…
  40. 90.Let f(x)=√2,x, x^2.
  41. 91.Let f(x)=casesax^2+2ax+34x^2+4x-3, & x -32,12[2mm] b, & x=-32,12cases be continuous at x=-…
  42. 92.Let f(x)=casesa|x|+x^2-2|x||x|x, & x0 b, & x=0cases be continuous at x=0.
  43. 93.Let ,R be such that the function f(x)=cases2(x^2-2)+2 x, & x<1 (+3)x+(-), & x1cases is dif…
  44. 94.Let [t] denote the greatest integer less than or equal to t.
  45. 95.Let y=y(x) be a differentiable function on (0,) such that y(1)=2, and [_t x(t^2y(x)-x^2y(t…
  46. 96.The value of [_x0_e[(e^x)(e^2x)(e^10x)]e^2-e^2 x] is equal to
  47. 97.Let f(x)=_0( x-x^2/(x-1)1+x^2/(x-1)), xR.
  48. 98.If _x2(x^3-5x^2+ax+b)(x-1-1)_e(x-1)=m, then a+b+m is equal to:
  49. 99.The product of all possible values of , for which [_x0((1-( x)((+1)x)((+2)x))/(^2((+1)x)))…
  50. 100.Let f(x) and g(x) be twice differentiable functions satisfying f''(x)=g''(x) for all xR, w…