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Limits, Continuity and Differentiability

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  1. 351.The value of [ _x 0 ( x 8 1 - x - 8 1 + x )] is equal to :
  2. 352.Let f : R [] R be a function such that f(2) = 4 and f'(2) = 1.
  3. 353.If [ = _x 4 ^3x - x ( x + 4 )] and [ = _x 0 ( x)^ x] are the roots of the equation, ax2 + …
  4. 354.Let f be any continuous function on [0, 2] and twice differentiable on (0, 2).
  5. 355.[ _x 0 ^2( ^4x ) x^4] is equal to :
  6. 356.[_x ()/(4) 8 √2-( x+ x)^7√2-√2 2 x] is equal to
  7. 357.If [ _n ( n^2 - n - 1 + n + ) = 0], then [8(+)] is equal to :
  8. 358.[ _x 2 ( ^2x( (2 ^2x + 3 x + 4)^1 2 - ( ^2x + 6 x + 2)^1 2 ) )] is equal to
  9. 359.Let f(x) be a polynomial function such that f(x) + f'(x) + f''(x) = x^5 + 64.
  10. 360.Let [= _x 0 x-(e^3 x-1) x(e^3 x-1)] for some [ R].
  11. 361.Let f : R [] R be a continuous function such that f(3x) - f(x) = x.
  12. 362.If [f(x) = pmatrix x + a & , & x 0 |x - 4| & , & x > 0 pmatrix .]
  13. 363.[ _x 0 ( x) - x x^4] is equal to :
  14. 364.[ _x 1 2 ( ^ - 1x) - x 1 - ( ^ - 1x)] is equal to :
  15. 365.Let a be an integer such that [ _x 7 18 - [1 - x] [x - 3a]] exists, where [t] is greatest …
  16. 366.The value of [ _n 6 _r = 1^n ^ - 1( 1 r^2 + 3r + 3 ) ] is equal to :
  17. 367.[ Let the function f(x)=[arraycl _e(1+5 x)- _e(1+ x)x & ; if x 0 10 & ; if x=0 array] be c…
  18. 368.If [_x 0 e^x+ e^-x+ xx ^2 x=(2)/(3)], where [, , R], then which of the following is NOT co…
  19. 369.The number of points, where the function [f: R R], [f(x)=|x-1| |x-2| |x-1|+(x-3)|x^2-5 x+4…
  20. 370.The value of [ _x 1 (x^2 - 1) ^2( x) x^4 - 2x^3 + 2x - 1] is equal to:
  21. 371.Let f and g
  22. 372.If [_x 0 e^a x- (b x)-cx e^-c x21- (2 x)=17], then 5 a^2+b^2 is equal to
  23. 373.The set of all values of a for which [ _x a ([x - 5] - [2x + 2]) = 0], where [[]] denotes …
  24. 374.[ _t 0 ( 1^1 ^2t + 2^1 ^2t, + ,..., + ,n^1 ^2t )^ ^2t] is equal to
  25. 375.The value of [ _n 1 + 2 - 3 + 4 + 5 - 6, + ,....., + ,(3n - 2) + (3n - 1) - 3n 2n^4 + 4n +…
  26. 376.Suppose [f: R (0, )] be a differentiable function such that [5 f(x+y)=f(x) f(y), x, y R].
  27. 377.[ _x (√3 x+1+√3 x-1)^6+(√3 x+1-√3 x-1)^6(x+x^2-1)^6+(x-x^2-1)^6 x^3 ]
  28. 378.If [ > > 0] are the roots of the equation a x^2+b x+1=0, and [_x (1)/()(1- (x^2+b x+a)2(1-…
  29. 379.[_x 0(((1- ^2(3 x). ^3(4 x))( ^3(4 x)( _e(2 x+1))^5))] is equal to _____________.
  30. 380.Let f: R R be defined as : [ f(x)= cases(a-b 2 x)/(x^2) ; & x1cases ] If f is continuous e…
  31. 381.Consider the function [f:(0,2) R] defined by [f(x)=(x)/(2)+(2)/(x)] and the function g(x)
  32. 382.[ If _x 0 (3+ x+ x+ _e(1-x))/(3 ^2 x)=(1)/(3) , then 2 - is equal to : ]
  33. 383.Consider the function.
  34. 384.[ _x 0 e^2| x|-2| x|-1x^2]
  35. 385.Let [f: R R] be a function given by [f(x)= cases(1- 2 x)/(x^2), & x 0cases] where [, R].
  36. 386.If the function [f(x)=( 3 x+ x- 3 x)/(x^3), x R], is continuous at x=0, then f(0) is equal…
  37. 387.[ _n ((1^2-1)(n-1)+(2^2-2)(n-2)++((n-1)^2-(n-1)) 1)/((1^3+2^3+ +n^3)-(1^2+2^2+ +n^2))] is …
  38. 388.For [a, b>0], let [f(x)= cases ((a+1) x)+b xx, & x 0cases] be a continuous function at x=0…
  39. 389.[ _x 0 e-(1+2 x)^(1)/(2 x)x] is equal to
  40. 390.If _x ((e1-e)(1e-(x)/(1+x)))^x=, then the value of _e 1+ _e equals :
  41. 391.If _r=1^n T_r=((2 n-1)(2 n+1)(2 n+3)(2 n+5))/(64), then _n _r=1^n((1)/(T_r)) is equal to :
  42. 392._x (2 x^2-3 x+5)(3 x-1)^(x)/(2)(3 x^2+5 x+4) (3 x+2)^x is equal to :
  43. 393.If the function [ f(x)=[gathered (2)/(x) (k_1+1) x+ (k_2-1) x, x0 gathered].
  44. 394._x 0 cosec x(√2 ^2 x+3 x- ^2 x+ x+4) is:
  45. 395.Let f: R-0 R be a function such that f(x)-6 f((1)/(x))=(35)/(3 x)-(5)/(2).
  46. 396.Let the function f(x)=(x^2-1)|x^2-a x+2|+ |x| be not differentiable at the two points x==2…
  47. 397.The value of _n (_k=1^n (k^3+6 k^2+11 k+5)/((k+3)!)) is :
  48. 398.For , , R, if _x 0 x^2 x+(-1) e^x^2 2 x- x=3, then +- is equal to :
  49. 399.Let f(x)= cases(1+a x)^1 / x & , x0cases be continuous at x=0.
  50. 400.Let f: R R be a continuous function satisfying f(0)=1 and f(2 x)-f(x)=x for all x R.