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Limits, Continuity and Differentiability

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  1. 1.For the function f(x)=e^ |x|-|x|,x R, consider the following statements : Statement I : f …
  2. 2.x 0lim(x^2 ^2xx^2- ^2x)
  3. 3.Let f(x)=y 0lim(1- (xy)) (xy)y^3.
  4. 4.Let f(x) and g(x) be twice differentiable functions satisfying f^''(x)=g^''(x) for all x R…
  5. 5.The product of all possible values of , for which x 0lim(1- ( x) (( +1)x) (( +2)x) ^2(( +1…
  6. 6.If x 2lim (x^3-5x^2+ax+b)(x-1-1) _e(x-1)=m, then a+b+m is equal to :
  7. 7.Let f(x)= 0lim( x-x^((2)/( )) (x-1)1+x^((2)/( ))(x-1)),x R.
  8. 8.Let y=y(x) be a differentiable function in the interval (0, ) such that y(1)=2, and _t x((…
  9. 9.Let [t] denote the greatest integer less than or equal to t
  10. 10.Let , R be such that the function f(x)= array|c|c| 2 (x^2-2)+2 x & ,x<1 ( +3)x+( - ) & ,x …
  11. 11.If the function f(x)=e^x(e^ x-x-1)+ _e( x+ x)-x x-x is continuous at x=0, then the value o…
  12. 12.If f(x)= array|c|c| a|x|+x^2-2( |x|)( |x|)x & ,x 0 b & ,x=0 array is continuous at x=0, th…
  13. 13.Let f(x)= array|c|c| ax^2+2ax+34x^2+4x-3 & ,x -(3)/(2),(1)/(2) b & ,x=-(3)/(2),(1)/(2) arr…
  14. 14.If x 0lime^(a-1)x+2 bx+(c-2)e^-xx x- _e(1+x)=2, then a^2+b^2+c^2 is equal to :
  15. 15.Let [ ] denote the greatest integer function, and let f(x)=min√2x,x^2.
  16. 16.Let f:R (0,) be a twice differentiable function such that f(3)=18,f^'(3)=0 and f^''(3)=4.
  17. 17.Given below are two statements: Statement I: x 0lim( ^-1x+ _e(1+x)/(1-x)-2xx^5)=(2)/(5) St…
  18. 18.x 0^+lim (5(x)^(1)/(3)) _e(1+3x^2)( ^-13x)^2(e^5(x)^(4)/(3)-1) is equal to
  19. 19.Let f be a differentiable function on R such that f(2)=1,f^'(2)=4.
  20. 20.Let f:R R be a continuous function satisfying f(0)=1 and f(2x)-f(x)=x for all x R.
  21. 21.If x 1^+lim(x-1)(6+ (x-1))+ (1-x)(x-1)^3=-1, where , R, then + is equal to
  22. 22.Let , f(x)=array|c|c| (1+ax)^1/x & ,x0 array be continuous at x=0.
  23. 23.If, x 0lim (2x)+a (4x)-bx^4is, finite,, then, (a+b), is, equal, to:
  24. 24.For , , R, if x 0limx^2 x+( -1)e^x^2 2x- x=3, then + - is equal to :
  25. 25.Let the function f(x)=(x^2-1)|x^2-ax+2|+ |x| be not differentiable at the two points x= =2…
  26. 26.The value of n lim( _k=1^nk^3+6k^2+11k+5(k+3)!) is :
  27. 27.Let [x] denote the greatest integer function, and let m and n respectively be the numbers …
  28. 28.x 0lim x(√2 ^2x+3 x- ^2x+ x+4) is:
  29. 29.Let f:R-0 R be a function such that f(x)-6f((1)/(x))=(35)/(3x)-(5)/(2).
  30. 30.x lim(2x^2-3x+5)(3x-1)^(x)/(2)(3x^2+5x+4)(3x+2)^x is equal to :
  31. 31.If the function f(x)=gathered (2)/(x) (k_1+1)x+ (k_2-1)x,, x0 gathered is continuous at x=…
  32. 32.If x lim(((e)/(1-e))((1)/(e)-(x)/(1+x)))^x= , then the value of _e 1+ _e equals :
  33. 33.If _r=1^nT_r=((2n-1)(2n+1)(2n+3)(2n+5))/(64), then n lim _r=1^n(1T_r) is equal to :
  34. 34.x 0lime-(1+2x)^(1)/(2x)x is equal to
  35. 35.For a,b>0, let f(x)=array|c|c| ( ((a+1)x)+b x)/(x), & x0 array be a continuous function at…
  36. 36.n lim(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 equal…
  37. 37.Let ,f:[-1,2] R be given by f(x)=2x^2+x+[x^2]-[x], where [t] denotes the greatest integer …
  38. 38.If the function f(x)= 3x+ x- 3xx^3,x R, is continuous at x=0, then f(0) is equal to :
  39. 39.If the function f(x)= array|c|c| 72^x-9^x-8^x+1√2-√1+ x, & x 0 a _e2 _e3 & ,x=0 array is c…
  40. 40.Let f:R R be a function given by f(x)=array|c|c| 1- 2xx^2, & x0 array where , R.
  41. 41.Let f(x)=|2x^2+5|x|-3|,x R.
  42. 42.Let f(x)= gathered x-1,x 2x,, x gatheredx N.
  43. 43.Let f:R R be defined as : f(x)=array|c|c| a-b 2xx^2; & x1 array If f is continuous everywh…
  44. 44.Consider the function f:(0,) R defined by f(x)=e^-| _ex|.
  45. 45.x 0lime^2| x|-2| x|-1x^2
  46. 46.Let g(x) be a linear function and f(x)= array|c|c| g(x) & ,x 0 ((1+x)/(2+x))^(1)/(x) & ,x>…
  47. 47.Consider the function f:(0,2) R defined by f(x)=(x)/(2)+(2)/(x) and the function g(x) defi…
  48. 48.x 0lim3+ x+ x+ _e(1-x)3 ^2x=(1)/(3)2 -
  49. 49.Consider the function.
  50. 50.If a=x 0lim√1+√1+x^4-√2x^4 and b=x 0lim ^2x√2-√1+ x, then the value of ab^3 is :