OCR Question Bank on Limit, Continuity, Differentiability PDF

QUESTION BANK ON LIMIT, CONTINUITY, DIFFERENTIABILITY OF FUNCTION  [STRAIGHT OBJECTIVE TYPE] Q.1 If both f (x) & g(x) are differentiable functions at x = x 0, then the function defined as, h(x) = Maximum {f(x), g(x)} (A) is always differentiable at x = x 0...

QUESTION BANK ON LIMIT, CONTINUITY, DIFFERENTIABILITY OF FUNCTION  [STRAIGHT OBJECTIVE TYPE] Q.1 If both f (x) & g(x) are differentiable functions at x = x 0, then the function defined as, h(x) = Maximum {f(x), g(x)} (A) is always differentiable at x = x 0 (B) is never differentiable at x = x 0 (C) is differentiable at x = x0 when f(x0)  g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x0). Q.2 If Lim (x3 sin 3x + ax2 + b) exists and is equal to zero then x0 (A) a =  3 & b = 9/2 (B) a = 3 & b = 9/2 (C) a =  3 & b = 9/2 (D) a = 3 & b = 9/2 x  x 1 Q.3 Let l = Lim   then {l}where {x}denotes the fractional part function is x   x  1  (A) 8 – e2 (B) 7 – e2 (C) e2 – 6 (D) e2 – 7  1 p if x  q where p & q  0 are relatively prime integers Q.4 For x > 0, let h(x) =  q  0 if x is irrational then which one does not hold good? (A) h(x) is discontinuous for all x in (0, ) (B) h(x) is continuous for each irrational in (0, ) (C) h(x) is discontinuous for each rational in (0, ) (D) h(x) is not derivable for all x in (0, ). Q.5 For a certain value of c, Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the x  value of the limit is (A) 1/5, 7/5 (B) 0, 1 (C) 1, 7/5 (D) none Q.6 If ,  are the roots of the quadratic equation ax 2 + bx + c = 0 then Lim  1  cos ax 2  bx  c  equals x  (x   ) 2 1   2 a2   2 a2   2 (A) 0 (B) (  ) (C) (  ) (D)  (  ) 2 2 2 Q.7 Lim x  3 ( xa )( xb) ( x  c) x =  a b  c (A) abc (B) (C) abc (D) (abc)1/3 3 Q.8 Lim x  tan 1 x  1  cot 1 x  2  is x   x2 x  1 1 (A) – 1 (B) (C) – (D) non existent 2 2  e x  cos 2 x  x Q.9 Given f (x) = for x  R – {0} x2 1 g (x) = f ({x}) for n < x < n + 2 where {x} denotes 1 = f (1 – {x} ) for n + < x < n + 1 , n I fractional part  2 function 5 = otherwise 2 then g (x) is (A) discontinuous at all integral values of x only (B) continuous everywhere except for x = 0 1 (C) discontinuous at x = n + ; n  I and at some x  I 2 (D) continuous everywhere Q.10 Let the function f, g and h be defined as follows 1 x sin   for  1  x  1 and x  0  x f (x) =   0 for x0 1 x 2 sin   for  1  x  1 and x  0  x g (x) =   0 for x0 h (x) = | x |3 for – 1  x  1 Which of these functions are differentiable at x = 0? (A) f and g only (B) f and h only (C) g and h only (D) none n   n  1 Q.11 Lim     sin  when   Q is equal to n    n  1  n   (A) e– (B) –  (C) e1 –  (D) e1 +  g (x ) Q.12 Let f (x) = h ( x ) , where g and h are cotinuous functions on the open interval (a, b). Which of the following statements is true for a < x < b? (A) f is continuous at all x for which x is not zero. (B) f is continuous at all x for which g (x) = 0 (C) f is continuous at all x for which g (x) is not equal to zero. (D) f is continuous at all x for which h (x) is not equal to zero.  2 cos x  sin 2 x e  cos x  1 Q.13 f (x) = ; g (x) = (  2 x ) 2 8 x  4 h (x) = f (x) for x < /2 = g (x) for x > /2 then which of the following holds? (A) h is continuous at x = /2 (B) h has an irremovable discontinuity at x = /2       (C) h has a removable discontinuity at x = /2 (D) f   = g   2  2    x  e x  cos 2x Q.14 If f(x) = , x  0 is continuous at x = 0, then x2 5 (A) f (0) = (B) [f(0)] = – 2 (C) {f(0)} = –0.5 (D) [f(0)]. {f(0)} = –1.5 2 where [x] and {x} denotes greatest integer and fractional part function x  b, x  0 Q.15 The function g (x) =  can be made differentiable at x = 0.  cos x , x  0 (A) if b is equal to zero (B) if b is not equal to zero (C) if b takes any real value (D) for no value of b Q.16 If f (x) = sin–1(sinx) ; x R then f is (A) continuous and differentiable for all x  (B) continuous for all x but not differentiable for all x = (2k + 1) , k I 2  (C) neither continuous nor differentiable for x = (2k – 1) ; k I 2 (D) neither continuous nor differentiable for x  R  [ 1,1] sin x Q.17 Limit where [ ] denotes greatest integer function , is  1  1  x 2 cos  (3 sin x  sin 3x ) 4  2 4 (A) (B) 1 (C) (D) does not exist      (1  cos x )  (1  cos x )  (1  cos x ) .........   1 Q.18 Lim  2  equals x 0 x 1 (A) 0 (B) (C) 1 (D) 2 2 x{x}  1 0  x 1 Q.19 Consider the function f (x) =  where {x} denotes the fractional part  2  {x} 1 x  2 function. Which one of the following statements is NOT correct? (A) Lim f ( x ) exists (B) f (0)  f (2) x1 (C) f (x) is continuous in [0, 2] (D) Rolles theorem is not applicable to f (x) in [0, 2]  x 2n  1 Q.20 The function f (x) = Lim is identical with the function n  x 2n  1 (A) g (x) = sgn(x – 1) (B) h (x) = sgn (tan–1x) (C) u (x) = sgn( | x | – 1) (D) v (x) = sgn (cot–1x) Q.21 Which one of the following statement is true? (A) If Lim f ( x ) ·g( x ) and Lim f ( x ) exist, then Lim g ( x ) exists. xc xc xc (B) If Lim f ( x ) ·g( x ) exists, then Lim f ( x ) and Lim g ( x ) exist. xc xc xc (C) If Lim f ( x )  g( x )  and Lim f ( x ) exist, then Lim g ( x ) exist. x c xc xc (D) If Lim f ( x )  g( x )  exists, then Lim f ( x ) and Lim g ( x ) exist. x c xc xc Q.22 The functions defined by f(x) = max {x 2, (x  1)2, 2x (1  x)}, 0  x  1 (A) is differentiable for all x (B) is differentiable for all x excetp at one point (C) is differentiable for all x except at two points (D) is not differentiable at more than two points. Q.23 Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable? |x| (A) f (x) = x1/3 (B) f (x) = (C) f (x) = e–x (D) f (x) = tan x x 2 2  (cos x  sin x )3  Q.24 The limiting value of the function f(x) = when x is 1  sin 2 x 4 1 3 (A) 2 (B) (C) 3 2 (D) 2 2 2 x  23  x  6  if x  2 x 1 x  2 2 Q.25 Let f (x) =  then   x 4 2  if x  2 x  3x  2 (A) f (2) = 8  f is continuous at x = 2 (B) f (2) = 16  f is continuous at x = 2 (C) f (2–)  f (2+)  f is discontinuous (D) f has a removable discontinuity at x = 2 Q.26 If Lim [ f ( x )  g ( x )]  2 and Lim [ f ( x )  g ( x )]  1 , then Lim f ( x ) g ( x ) x a x a x a 3 3 4 (A) need not exist (B) exist and is (C) exists and is – (D) exists and is 4 4 3 sin 2 ( x 3  x 2  x  3) Q.27 Lim has the value equal to x 1 1  cos( x 2  4 x  3) (A) 18 (B) 9/2 (C) 9 (D) none Q.28 The graph of function f contains the point P (1, 2) and Q(s, r). The equation of the secant line  s 2  2s  3  through P and Q is y =  s  1  x – 1 – s. The value of f ' (1), is   (A) 2 (B) 3 (C) 4 (D) non existent Q.29 Consider f(x) =     2 sin x  sin 3 x  sin x  sin 3 x   , x  for x  (0, )     2 sin x  sin 3 x  sin x  sin 3 x   2 f(/2) = 3 where [ ] denotes the greatest integer function then, (A) f is continuous & differentiable at x = /2 (B) f is continuous but not differentiable at x = /2 (C) f is neither continuous nor differentiable at x = /2 (D) none of these Q.30 Let [x] denote the integral part of x  R. g(x) = x  [x]. Let f(x) be any continuous function with f(0) = f(1) then the function h(x) = f(g(x)) (A) has finitely many discontinuities (B) is discontinuous at some x = c (C) is continuous on R (D) is a constant function. Q.31 If f (x + y) = f (x) + f (y) + | x | y + xy2,  x, y  R and f ' (0) = 0, then (A) f need not be differentiable at every non zero x (B) f is differentiable for all x  R (C) f is twice differentiable at x = 0 (D) none 1 Q.32 Lim (lnx ) e x is x e  1 (A) ee (B) ee (C) e–2 (D) e–1 12 n  2 2 (n1)32 (n2)..... n 2.1 Q.33 Lim is equal to : n  13  23 33 ...... n 3 1 2 1 1 (A) (B) (C) (D) 3 3 2 6 Q.34 Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true? I. f is continuous on the closed interval [a, b] II. f is bounded on the open interval (a, b) III. If a 0 let f (x) =  (Q) e2  3 ln (a  x )  2, if x  0 If f is continuous at x = 0 then 'a' equals (R) 1/e xx  aa xx  ax (C) Let L = Lim and M = Lim (a > 0). x a x a x a x a If L = 2M then the value of 'a' is equal to (S) non existent  [STRAIGHT OBJECTIVE TYPE] Q.1 C Q.2 A Q.3 D Q.4 A Q.5 A Q.6 C Q.7 B Q.8 B Q.9 D Q.10 C Q.11 C Q.12 D Q.13 B Q.14 D Q.15 D Q.16 B Q.17 A Q.18 B Q.19 C Q.20 C Q.21 C Q.22 C Q.23 A Q.24 D Q.25 C Q.26 B Q.27 A Q.28 C Q.29 A Q.30 C Q.31 B Q.32 A Q.33 A Q.34 D Q.35 A Q.36 B Q.37 C Q.38 C Q.39 A Q.40 C Q.41 D Q.42 A Q.43 D Q.44 D Q.45 B Q.46 A Q.47 C Q.48 A Q.49 A Q.50 B Q.51 A Q.52 C Q.53 C Q.54 D Q.55 D Q.56 D Q.57 D Q.58 C Q.59 D Q.60 B Q.61 C Q.62 B Q.63 B Q.64 D Q.65 A Q.66 C Q.67 B Q.68 C Q.69 B Q.70 B Q.71 A Q.72 C Q.73 C Q.74 A Q.75 D [MULTIPLE OBJECTIVE TYPE] Q.76 B,C Q.77 A,C Q.78 A,C Q.79 A,B,D Q.80 A,B,C Q.81 B,C,D Q.82 A, C, D Q.83 A, B, D Q.84 A,B,C,D Q.85 B,D Q.86 A,B Q.87 A,B,D Q.88 A,C Q.89 B,C Q.90 A,D Q.91 A,C Q.92 B,C [MATCH THE COLUMN] Q.93 (A) S; (B) R; (C) P; (D) Q ; (E) P Q.94 (A) R, S; (B) P, S; (C) P, Q; (D) P, S Q.95 (A) S; (B) R; (C) Q; (D) P Q.96 (A) P, S; (B) P, Q; (C) R, S; (D) P, Q Q.97 (A) S; (B) P ; (C) Q; (D) R Q.98 (A) R; (B) S; (C) P; (D) Q Q.99 (A) Q; (B) R; (C) P; (D) P Q.100 (A) S ; (B) P, Q ; (C) P ANSWER KEY