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Question bank on function limit continuity & derivability There are 105 questions in this question bank. Select the correct alternative : (Only one is correct) Q.13 If both f(x) & g(x) are differentiable functions at x = x0, then the function defined as, h(x) = Maximum {f(x), g(x)} : (A) is always differentiable at x = x0 (B) is never differentiable at x = x0 (C*) is differentiable at x = x0 provided f(x0) g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x0) . [Hint: Consider the graph of h(x) = max(x, x2) at x = 0 and x = 1] Q.25 If Lim (x3 sin 3x + ax2 + 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 Lim sin 3x + a + b Lim sin 3x + ax + bx3 [Sol. x0 x3 x2 = x0 x3 3 sin 3x + a + bx 2 = Lim 3x for existence of limit 3 + a = 0 a = – 3 x0 x2 Lim sin 3x 3x + bx3 27. sin t t + b l = x0 x3 = t3 = 0 (3x = t) = 27 + b 6 = 0 b = 9 ] 2 [ OR use L' Hospital's rule ] xm sin 1 x 0, m N Q.36 A function f(x) is defined as f(x) = x 0 continuous at x = 0 is if x = 0 . The least value of m for which f (x) is (A) 1 (B) 2 (C*) 3 (D) none hm sin 1 [Sol. f ' (0+) = Lim x must exist h0 h m > 1 m xm1 sin 1 xm2 cos 1 x 0 for m > 1 h ' (x) = 0 if x x x = 0 now Lim h(x) = Lim m hm1 sin 1 hm2 cos 1 h0 limit exist if m > 2 h0 h h m N m = 3] 1 if x=p where p & q > 0 are relatively prime integers Q.410 For x > 0, let h(x) = q 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, ) . 2 h 2 = 1 [Sol. Let x = 3 which is rational 3 3 Lim hFG2 + tJ= 0 discontinuous at x Q t0 Let x = H3 K / Q h d2 i = 0 consider hd2 i = h FG1.414023583J= 1 = 1.41401235839 0 H 1010 1010 Hence h is continuous for all irrational A ] 1 1 2xn ex 3xn ex Q.5 The value of Limit (where nN ) is 12 2 x xn 2 (A) ln 3 (B*) 0 (C) n ln 3 (D) not defined xn xn [Hint: l = Limit 2 ex – 3 ex n xn but Limit x = 0 l = 0 ] x x x e Q.613 For a certain value of c, of the limit is Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value x (A*) 1/5, 7/5 (B) 0, 1 (C) 1, 7/5 (D) none 7 2 c 5c1 7 2 c [Sol. Lim x5c 1+ + x = Lim x x 1+ + 1 x x x5 x x x5 This will be of the form × 0 only if 5c – 1 = 0 c = 1/ 5 substituting c = 1/5 , l becomes Hence l = Lim x x[(1+ X)1/ 5 – 1] where X = 7 x + 2 x5 Lim x 1+ X +..... 1 Lim x

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- Question bank on function limit continuity & derivability There are 105 questions in this question bank. Select the correct alternative : (Only one is correct) Q.13 If both f(x) & g(x) are differentiable functions at x = x0, then the function defined as, h(x)=Maximum{f(x),g(x)}: (A) is always differentiable at x = x0 (B) is never differentiable at x = x0 (C*) is differentiable at x = x0 provided f(x0) g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x0) . [Hint: Consider the graph of h(x) = max(x, x2) at x = 0 and x = 1] Q.25 If 0 x Lim (x3 sin 3x + ax2 + b) exists and is equal to zero then : (A*) a = 3 & b = 9/2 (B) a = 3 & b = 9/2 (C) a = 3 & b = 9/2 (D) a = 3 & b = 9/2 [Sol. b x a x x 3 sin Lim 2 3 0 x = 0 x Lim 3 3 x bx ax x 3 sin = 0 x Lim 2 2 x bx a x 3 x 3 sin 3 for existence of limit 3 + a = 0 a = – 3 l = 0 x Lim 3 3 x bx x 3 x 3 sin = b t t t sin . 27 3 = 0 (3x = t) = b 6 27 = 0 b = 2 9 ] [ OR use L' Hospital's rule ] Q.36 A function f(x) is defined as f(x) = x x m N if x m x sin , 1 0 0 0 . Theleast value of m forwhich f(x)is continuous at x =0 is (A) 1 (B) 2 (C*) 3 (D) none [Sol. f ' (0+) = h x 1 sin h Lim m 0 h must exist m > 1 for m > 1 h ' (x) = 0 x if 0 0 x x 1 cos x x 1 sin x m 2 m 1 m now h 1 cos h h 1 sin h m Lim ) x ( h Lim 2 m 1 m 0 h 0 h limit existif m>2 m N m = 3]
- Q.410 For x > 0, let h(x) = integers prime relatively are 0 q & p where irrational is x if 0 x if q p q 1 then which onedoes not hold good? (A*) h(x)isdiscontinuous for all xin (0,) (B)h(x)iscontinuous for eachirrationalin (0, ) (C)h(x)isdiscontinuous for each rationalin (0, ) (D) h(x) is not derivable for all x in (0, ). [Sol. Let x = 2 3 which is rational 3 1 3 2 h Lim h t t F H G I K J 0 2 3 0 discontinuous at x Q Let x = 2 Q h 2 d i = 0 consider 2 = 1.41401235839 h 2 d i = h 1414023583 1010 . F H G I K J= 1 1010 0 Hencehiscontinuous forallirrational A A ] Q.512 The value of n e 1 x e 1 x x x 3 2 Limit x n x n (where N n ) is (A) ln 3 2 (B*) 0 (C) n ln 3 2 (D)not defined [Hint: l = n e x e x x x 3 2 Limit x n x n but 0 e x Limit x n x l = 0 ] Q.613 For a certain value of c, x Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value ofthelimitis (A*) 1/5, 7/5 (B) 0, 1 (C) 1, 7/5 (D) none [Sol. x Lim x x 2 x 7 1 x c 5 c 5 = x Lim 1 x 2 x 7 1 x x c 5 1 c 5 This will beof theform ×0onlyif 5c – 1 = 0 c = 1/ 5 substituting c = 1/5 , l becomes Hence l = x Lim 1 X 1 x 5 / 1 where 5 x 2 x 7 X = x Lim 1 ..... 5 X 1 x = x Lim 5 1 . x 2 x 7 x 5 = 5 7 Hence c = 5 1 and l = 5 7 ]
- Q.714 Considerthepiecewisedefinedfunction f (x) = 4 x if 4 x 4 x 0 if 0 0 x if x choose theanswerwhichbest describes thecontinuityofthis function (A)The functionis unbounded and thereforecannot be continuous. (B)The functionis right continuous atx =0 (C) Thefunction has a removablediscontinuityat 0 and4, but is continuous on the rest ofthe real line. (D*)Thefunctionis continuous ontheentirereal line [Hint: Refer tothe graph ] Q.817 If , arethe roots of thequadraticequation ax2 +bx+ c= 0 then x Lim 1 2 2 cos ( ) ax bx c x equals (A) 0 (B) 1 2 ()2 (C*) a2 2 ()2 (D) a2 2 ()2 [Sol. Lim ax bx c ax bx c ax bx c x x F H G I K J 0 2 2 2 2 2 2 2 2 2 2 4 sin ( ) c h c h c h ; Lim a x b a x c a x b a x c a x x x F H G I K J F H G I K J 2 4 2 2 2 ( ) ( ) Lim a x x x x 2 2 2 2 2 ( ) ( ) ( ) ; Lim a C x 2 2 2 ( ) ] Q.918 Which one ofthe followingbest representsthe graph ofthefunctionf(x)= nx tan 2 Lim 1 n (A*) (B) (C) (D) Q.1019 1 x Lim 4 1 3 1 3 1 2 1 2 3 1 4 3 1 x x x x x x x x . = (A) 1 3 (B*) 3 (C) 1 2 (D) none
- Q.1124 ABCis anisoscelestriangleinscribedinacircleofradiusr.IfAB=AC &histhealtitudefromAtoBC and Pbe the perimeter ofABC then 0 h Lim 3 P equals (where is the area of the triangle) (A) r 32 1 (B) r 64 1 (C*) r 128 1 (D) none [Sol. AB = 2 2 h rh 2 h = rh 2 P = rh 2 2 + 2 h rh 2 2 P = 2 h rh 2 rh 2 2 = 2 h . h rh 2 2 2 = 2 h rh 2 h 3 0 h p Limit = 3 2 2 0 h h rh 2 rh 2 8 h rh 2 h Limit = 3 2 / 3 2 / 3 0 h h r 2 r 2 h 8 h r 2 h Limit = r 128 1 ) r 2 ( ) r 2 ( 8 . 8 r 2 2 / 1 (C) ] Alternative: Note that as h 0 , b = 2 a or 2b = a Hence 3 p = 3 ) b 2 a ( . R 4 ) b ( ab (using R = 4 abc ) = 3 3 b 64 R 4 b 2 = R 128 1 (C) ] Q.1232 Let the function f, g and h be defined as follows : f (x) = 0 x for 0 0 x and 1 x 1 for x 1 sin x g (x) = 0 x for 0 0 x and 1 x 1 for x 1 sin x2 h (x) = | x |3 for – 1 x 1 Which ofthesefunctions aredifferentiableatx =0? (A) f and g only (B) f and h only (C*) g and h only (D) none Q.1333 If [x] denotes the greatest integer x, then Limit n 1 4 n 1 2 3 3 3 x x n x ...... equals (A) x/2 (B) x/3 (C) x/6 (D*) x/4
- Q.1436 Let f (x) = ) x ( ) x ( h g , where g and h are cotinuous functions on the open interval (a, b). Which of the followingstatements 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. [Hint: Bytheorem,ifgandharecontinuous functions ontheopen interval (a,b),then g/h is alsocontinuous at all x in the open interval (a, b) where h(x) is not equal to zero. ] Q.1539 The period of thefunction f (x) = | x cos x sin | | x cos | | x sin | is (A) /2 (B) /4 (C*) (D) 2 Q.1645 If f(x) = 2 x x x 2 cos e x , x 0 is continuous at x = 0, then (A) f (0) = 2 5 (B) [f(0)] = – 2 (C) {f(0)} = –0.5 (D*) [f(0)] . {f(0)} = –1.5 where [x] and{x} denotes greatest integerand fractional part function [Hint: 2 x 0 x x ) x 2 cos 1 ( 1 e x Lim = – 2 1 – 2 = – 2 5 ; hence for continuity f (0) = – 2 5 [f (x)] = – 3 ; {f (0)} = 2 5 = 2 1 ; hence [f (0)] {f (0)} = – 2 3 = – 1.5 ] Q.1746 The valueof thelimit 2 n 2 n 1 1 is (A) 1 (B) 4 1 (C) 3 1 (D*) 2 1 [Hint: 2 n 2 2 n 1 n = 2 n 2 n n 1 n n 1 n = n · ) 1 n ......( 4 · 3 · 2 ) 1 n ........( 3 · 2 · 1 = 2 1 n · n 1 = 2 n 1 1 Lim n = 2 1 ] Q.1847 The function g (x) = 0 x , x cos 0 x , b x can be made differentiable at x = 0. (A) if b is equal to zero (B) if b is not equal to zero (C) if b takes anyreal value (D*) for no value of b
- [Hint: g' (0+) = h 1 cosh Lim 0 h = 0 g ' (0–) = h 1 b h Lim 0 h for existence of line b = 1 thus g ' (0–) = 1 Hence gcan not be madedifferentiable for anyvalueof b.] Q.1949 Let f be differentiable at x = 0 and f ' (0) = 1. Then h ) h 2 ( ) h ( Lim 0 h f f = (A*) 3 (B) 2 (C) 1 (D) – 1 [Hint: reduces to 3 f ' (0) ] Q.2050 If f (x) = sin–1(sinx) ; R x then f is (A)continuousanddifferentiableforallx (B*) continuous for all x but not differentiable for all x = (2k + 1) 2 , I k (C) neither continuous nor differentiable for x = (2k – 1) 2 ; I k (D)neithercontinuousnordifferentiablefor ] 1 , 1 [ R x [Hint: 2 3 k 2 x 2 k 2 for x ) 1 k 2 ( 2 k 2 x 2 k 2 for k 2 x ] Q.2153 ) x 3 sin x sin 3 ( 4 1 cos x sin Limit 1 2 x where [ ] denotes greatest integer function , is (A*) 2 (B) 1 (C) 4 (D) doesnot exist [Hint: ] x [sin cos x sin Limit 3 1 2 x ax x /2 , [sin3x] 0 and sinx 1 l = 2 (A) ] Q.2260 If 0 x Lim x ) x 3 ( n ) x 3 ( n l l = k , the value of k is (A*) 3 2 (B) – 3 1 (C) – 3 2 (D) 0
- [Sol. 0 x Limit x ) x 3 ( n ) x 3 ( n l l = 0 x Limit x 3 x 1 n 3 n 3 x 1 n 3 n l l l l = 0 x Limit x / 1 x / 1 3 x 1 n 3 x 1 n l l = 3 x x 1 3 x . x 1 = 3 2 3 1 3 1 (A) ] Q.2364 The function f (x) = 1 x 1 x Lim n 2 n 2 n isidentical withthefunction (A) g (x) = sgn(x – 1) (B) h (x) = sgn (tan–1x) (C*) u (x) = sgn( | x | – 1) (D) v (x) = sgn (cot–1x) [Hint: f (x) = 1 | x | if 1 1 or 1 x if 0 1 x 1 if 1 and g (x) = sgn(|x| – 1) is same ] Q.2469 The functions defined by f(x) = max {x2, (x 1)2, 2x (1 x)}, 0 x 1 (A) isdifferentiableforallx (B) is differentiableforall x excetp at one point (C*) is differentiable for all x except at two points (D) is not differentiable at morethan two points. Q.2573 f (x) = nx x l and g (x) = x nx l . Then identifythe CORRECT statement (A*) ) x ( g 1 andf (x) areidentical functions (B) ) x ( f 1 and g(x)are identical functions (C) f (x) . g (x) = 1 0 x (D) 1 ) x ( g . ) x ( f 1 0 x [Hint: (A) nx x ) x ( f ; x / nx 1 ) x ( g 1 l l x > 0 , x 1 for both (B) x nx ) x ( g ; nx / x 1 ) x ( f 1 l l ) x ( f 1 is not defined at x =1 but g (1) = 0 (C) f (x) . g (x) = x nx . nx x l l = 1 if x > 0 , x 1 N. I. (D) 1 x nx . nx x 1 ) x ( g . ) x ( f 1 l l only for x > 0 and x 1 ] Q.2674 If f(3) = 6 & f(3) = 2, then Limit x 3 x f f x x ( ) ( ) 3 3 3 is given by : (A) 6 (B) 4 (C*) 0 (D) none of these
- [Sol. f (3) = 6 ; f (3)= 2 Limit x 3 x f f x x ( ) ( ) 3 3 3 , put x = 3 + h = h ) h 3 ( f 3 ) 3 ( f · ) h 3 ( Limit 0 h = ) 3 ( f h ] ) 3 ( f ) h 3 ( f [ 3 Limit 0 h = – 3f (3) + f (3) = – 6 + 6 = 0 Ans ] Q.2776 Which oneof the followingfunctionsis continuous everywhere inits domain but has atleast onepoint whereitisnotdifferentiable? (A*) f (x) = x1/3 (B) f (x) = x | x | (C) f (x) = e–x (D) f (x) = tan x [Hint: x1/3 is not differentiable at x = 0] Q.2886 The limitingvalue of thefunction f(x) = x 2 sin 1 ) x sin x (cos 2 2 3 when x 4 is (A) 2 (B) 1 2 (C) 3 2 (D*) 3 2 [Hint: put x = 4 + h or use L'Hospital's rule ] Q.2989 Let f (x) = 2 x if 2 x 3 x 4 x 2 x if 2 2 6 2 2 2 x 1 x x 3 x then (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 discontinuityat x = 2 [Hint: f (2+) = 8 ; f (2–) = 16 ] Q.3090 On the interval I = [2, 2], the function f(x) = ( ) ( ) ( ) | | x e x x x x 1 0 0 0 1 1 then which oneof the followingdoesnot holdgood? (A*) is continuous for all values of x I (B) is continuous for x I (0) (C) assumes all intermediate values from f(2) & f(2) (D) has amaximum value equal to 3/e. [Sol. f (x) = 0 x if 0 0 x if 1 x 0 x if e ) 1 x ( x / 2 the graph off (x) is hence f can assume all values for f (– 2) to f (2) ]
- Q.3193 Whichofthefollowingfunctionissurjectivebutnotinjective (A) f : R R f (x) = x4 + 2x3 – x2 + 1 (B) f : R R f (x) = x3 + x + 1 (C) f : R R+ f (x) = 2 x 1 (D*) f : R R f (x) = x3 + 2x2 – x + 1 [Sol. (A) f (x) = x4 + 2x3 – x2 + 1 Apolynomial of degree even will always be into say f (x) = a0x2n + a1 x2n–1 + a2 x2n – 2 + .... + a2n x Limit f (x) = x Limit [x2n n 2 n 2 2 2 1 0 x a .... x a x a a = 0 a if 0 a if 0 0 Hence it will never approach / – (B) f (x) = x3 + x + 1 f (x) = 3x2 + 1 – injective as well as surjective (C) f (x) = 2 x 1 – neitherinjectivenor surjective f (x) = x3 + 2x2 – x + 1 f (x) =3x2 + 4x – 1 D > 0 Hencef(x) is surjectivebutnot injective.] Q.3294 Consider the function f (x) = 3 x 2 if x 6 2 x if 1 2 x 1 if ] x [ x where [x] denotes step up functionthen at x =2 function (A)has missingpointremovablediscontinuity (B*)hasisolatedpointremovablediscontinuity (C)has nonremovablediscontinuityfinitetype (D)iscontinuous [Hint: fromthefigurethefunctionhasanobvious removableisolatedpointdiscontinuous. Q.3395 Suppose that f is continuous on [a, b] and that f (x) is an integer for each x in [a, b]. Then in [a, b] (A) f is injective (B) Range of f mayhavemanyelements (C) {x} is zero for all x [a, b] where { } denotes fractional part function (D*) f(x) is constant [Hint: Explanation : Let f (x1) = n and f (x2) = m, x1, x2 (a, b) with n > m (say). According to the intermediate value theorem, between x1 and x2 there must be some value x for which f (x) = m + 2 1 which is impossiblesince m + 2 1 is not aninteger.] Q.34101 Thegraphof function f contains the point P (1, 2) andQ(s, r). Theequation ofthe secant line through P and Q is y = 1 s 3 s 2 s2 x – 1 – s. The value of f ' (1), is (A) 2 (B) 3 (C*) 4 (D)non existent
- [Sol. I Bydefinition f '(1) is the limit of the slope of the secant line when s 1. Thus f '(1) = 1 s 3 s 2 s Lim 2 1 s = 1 s ) 3 s )( 1 s ( Lim 1 s = ) 3 s ( Lim 1 s = 4 (D) II Bysubstitutingx = s intothe equation of thesecant line, and cancellingbys – 1 again, we get y = s2 + 2s – 1. This is f (s), and its derivative is f '(s) = 2s + 2, so f ' (1) = 4.] Q.35108 The range of the function f(x) = 12 x 11 x 2 ) 10 x 7 x ( 5 x n e 2 2 ) 2 x ( x 2 l is (A*) ) , ( (B) ) , 0 [ (C) , 2 3 (D) 4 , 2 3 [Sol. f (x) = ) 4 x ( ) 3 x 2 ( ) 5 x ( ) 2 x ( . 5 nx e ) 2 x ( x 2 l Note that at x = 3/2 & x = 4function is not defined andin open interval (3/2,4)function is continuous. ) ve ( ) ve ( ) ve ( ) ve ( ) ve ( Lim 2 3 x ) ve ( ) ve ( ) ve ( ) ve ( ) ve ( Lim 4 x In the openinterval (3/2,4) the function iscontinuous & takes up all real values from (– ,) Hence range of the function is (– , ) or R] Q.36113 Consider f(x) = 2 2 3 3 3 3 sin sin sin sin sin sin sin sin x x x x x x x x , x 2 for x (0, ) 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 [Sol. In the immediate neighborhood of x = /2 , sinx > sin3x |sinx – sin3x|= sinx – sin3x Hence for x /2 , f (x) = x sin x sin ) x sin x (sin 2 x sin x sin ) x sin x (sin 2 3 3 3 3 = 3 x sin x sin x sin 3 x sin 3 3 3 Hence f is continuous and diff. at x = /2 ] Q.37114 The number of points at which the function, f(x) = x – 0.5 + x – 1 + tan x does not have a derivativeintheinterval (0,2) is : (A) 1 (B) 2 (C*) 3 (D) 4
- Q.38117 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) hasfinitelymanydiscontinuities (B) is discontinuous at some x = c (C*) is continuous on R (D) is a constant function . [Sol. g(x) = x – [x] = {x} fis continuouswithf(0)=f(1) h(x) = f (g(x)) = f ({x}) Let the graphof f is as shownin the figure satisfying f(0) = f(1) now h(0) = f ({0}) = f(0) = f(1) h(0.2) = f ({0.2}) = f(0.2) h(1.5) = f ({1.5}) = f(0.5) etc. Hence the graphof h(x) will be periodicgraph as shown h is continuous in R C ] Q.39118 Given the function f(x) = 2x x3 1 + 5 x 1 4 x + 7x2 x 1 + 3x + 2 then : (A) thefunction is continuous but not differentiable at x = 1 (B*) thefunction is discontinuous at x = 1 (C) the function is both cont. & differentiable at x = 1 (D) the range of f(x) is R+. Q.40119 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 everynon zero x (B*) f is differentiablefor all x R (C) f is twice differentiable at x = 0 (D) none [Hint: f '(x) = h xh h | x | ) h ( Lim h ) x ( ) h x ( Lim 2 0 h 0 h f f f f (0) = 0 f ' (x) = xh | x | h ) 0 ( ) h ( Lim 0 h f f f ' (x) = f ' (0) + | x | = | x | ] Q.41131 For } x 10 { } 10 x sin{ Lim 8 x (where { } denotes fractional part function) (A)LHLexist but RHLdoes not exist (B*) RHLexist but LHLdoes not exist. (C) neitherLHLnor RHLdoes not exist (D) both RHLand LHLexist and equals to 1
- [Hint: } x { } x sin{ Lim 8 x = 0 as {I + x } = {x} ; as 0 } x { Lim I x and 1 } x { Lim I x } x { } x sin{ Lim 8 x as sin{x} sin(1) and {–x} 0 ] Q.42134 n Lim 3 3 3 3 2 2 2 2 n ...... 3 2 1 1 . n ..... ) 2 n ( 3 ) 1 n ( 2 n 1 is equal to : (A*) 1 3 (B) 2 3 (C) 1 2 (D) 1 6 [Sol. Lim n n n n n n n n 1 2 1 3 2 1 2 2 2 2 3 ( ) ( ) ............ ( ( )) Nr. = n n n n ( ....... ) ( . . . .......... ( ) ) 1 2 12 2 3 34 1 2 2 2 2 2 2 2 = n n r r r n 2 2 2 1 ( ). = n n r r r n 2 3 2 1 ( ) = n n n n 2 3 2 = ( ) n n n 1 2 3 Lim n n n n n ( ) 1 2 3 3 Lim n n n n n n n n n ( ) ( )( ) ( ) ( ) 1 1 2 1 6 1 1 1 4 3 1 1 3 A Alternatively:l = 1 n ...... 3 2 1 1 · n ...... ) 1 n ( 2 n 1 3 2 3 3 2 2 2 – 1 = 3 2 2 2 n ) 1 n ( n ...... ) 1 n ( 2 ) 1 n ( 1 – 1 l = 3 2 n n ) 1 n ( – 1 ] Q.43135 The domain of definition of the function f (x) = | 6 x x | log 2 x 1 x + 16–xC2x–1 + 20–3xP2x–5 is (A*) {2} (B) } 3 , 2 { , 4 3 (C) {2, 3} (D) , 4 1 Where [x] denotesgreatest integerfunction. Q.44136 If f (x) = 10 x 7 x 25 bx x 2 2 for x 5 and f is continuous at x = 5, then f (5) has the value equal to (A*) 0 (B) 5 (C) 10 (D) 25
- [Hint: f (x) = ) 5 x )( 2 x ( 25 bx x2 . for existence of limit 25 – 5b + 25 = 0 b = 10 now f (x) = ) 5 x )( 2 x ( ) 5 x )( 5 x ( and ) x ( f Lim 5 x = 0 (A) ] Q.45139 Let f beadifferentiablefunctionontheopen interval(a,b). Whichof thefollowing statementsmustbe true? I. f is continuous on the closedinterval [a, b] II. f is bounded on the open interval (a, b) III. If a<a1<b1<b, and f (a1)<0< f (b1), then there is a number c such that a1<c< b1 and f (c)=0 (A) I and II only (B) I and III only (C) II and III only (D*) onlyIII [Sol. I and IIare false. The function f (x) = 1/x, 0 < x <1, is a counter example. Statement IIIis true.Applythe intermediate valuetheorem to f on the closed interval [a1, b1] ] Q.46145 The value of a log a sec x log x cot x x 1 a a 1 x Limit (a > 1) is equal to (A*) 1 (B) 0 (C) /2 (D) does not exist [Hint: x log a sec x x log cot Limit a x 1 a a 1 x ; 0 x x log as a a x and x log a a x (using L’opital rule) l = 2 / 2 / = 1 ] Q.47148 Let f : (1, 2 ) R satisfies the inequality 2 x | 8 x 4 | x ) x ( f 2 33 ) 4 x 2 cos( 2 , ) 2 , 1 ( x . Then ) x ( f Lim 2 x is equal to (A) 16 (B*) –16 (C)cannotbedeterminedfromthegiven information (D) doesnot exists [Hint: ) x ( f Lim 2 x 16 2 33 ) 4 x 2 cos( ; ) x ( f Lim 2 x 16 ) 4 ( 4 2 x | 2 x | 4 . x2 Bysandwich theorem ) x ( f Lim 2 x = –16 ]
- Q.48163 Let a = min [x2 + 2x + 3, x R] and b = x x 0 x e e x cos x sin Lim . Then the value of n 0 r r n r b a is (A) n 1 n 2 · 3 1 2 (B) n 1 n 2 · 3 1 2 (C) n n 2 · 3 1 2 (D*) n 1 n 2 · 3 1 4 [Sol. a = (x + 1)2 + 2 a = 2 b = x 2 x 2 ). 1 e ( 2 x 2 sin Lim x 2 0 x = 2 1 now n 0 r r n r b a = r n r 2 1 2 = n 0 4 r 2 n 2 2 1 = n 2 1 [1 + 22 + 24 + ... + 22n ] = n 2 1 1 2 1 ) 2 ( 2 1 n 2 = n 2 1 3 1 4 1 n = n 1 n 2 . 3 1 4 ] Q.49165 Period of f(x) = nx + n [nx + n], (n N where [ ] denotes the greatest integer function is : (A) 1 (B*) 1/n (C) n (D) none of these Q.50171 Let f beareal valued functiondefined by f(x)=sin1 1 3 x +cos1 x 3 5 .Then domain of f(x) isgivenby: (A*) [ 4, 4] (B) [0, 4] (C) [ 3, 3] (D) [ 5, 5] Q.51172 For the functionf (x) = ) x ( sin n 1 1 Lim 2 n , which of thefollowing holds? (A) The range of f is a singleton set (B)f iscontinuous on R (C*) f is discontinuous for all x I (D) f is discontinuous for some x R [Hint: f (x) = I x if 0 I x if 1 fis discontinuous for all x I ] Q.52174 Domain of the function f(x) = x cot n 1 1 l is (A) (cot1 , ) (B) R – {cot1} (C) (– ,0) (0,cot1) (D*) (– , cot1) [Sol. ln (cot–1x) > 0 cot–1x > 1 x < cot 1 domain (– , cot1) Ans. ]
- Q.53175 The function Q x , 5 x 2 x Q x , 1 x 2 ) x ( f 2 is (A)continuousnowhere (B)differentiablenowhere (C)continuousbut not differentiableexactlyatonepoint (D*)differentiableandcontinuous onlyatonepointand discontinuouselsewhere [Hint: Diff. & cont. onlyat x = 2 which is the point of intersection of y= 2x + 1 and y= x2–2x +5 ] [Sol. y = (x – 1)2 + 4 = x2 – 2x + 5 = 2x + 1 = x2 – 4x + 4 = 0 = (x – 2)2 = 0 x = 2 line is tangent at x = 2 ] Q.54180 For the function f (x) = ) 2 x ( 1 2 x 1 , x 2 which of the following holds? (A) f (2) = 1/2 and f is continuous at x =2 (B) f (2) 0, 1/2 and f is continuous at x = 2 (C*) f can not be continuous at x = 2 (D) f (2) = 0 and f is continuous at x = 2. Q.55184 ) x tan(sin 1 ) x cos(sin x Lim 1 1 2 1 x is (A) 2 1 (B*) – 2 1 (C) 2 (D) – 2 [Hint: put sin–1x = tan 1 cos sin Lim 4 = cos Lim 4 = – 2 1 ] Q.56186 Which oneof the followingis not bounded on theintervals as indicated (A) f(x) = 1 x 1 2 on (0, 1) (B*) g(x) = x cos 1 x on (–) (C) h(x) = xe–x on (0, ) (D) l (x) = arc tan2x on (–, ) [Sol. (A) 0 x Limit f (x) = 2 1 2 Limit 1 h 1 0 h ; 1 x Limit f (x) = 0 2 Limit h 1 0 h 2 1 , 0 ) x ( f bounded (C) 0 h Limit x e–x = 0 h Limit h e –h = 0 ; x Limit x e–x = x Limit 0 e x x Also x e x y y = x 2 x x e xe e e x (1 – x) e 1 , 0 ) x ( h ]
- Q.57190 The domainof thefunction f(x)= arc x x x cot 2 2 , where [x] denotesthe greatest integernotgreater than x,is: (A) R (B) R {0} (C*) R n n I : { } 0 (D) R {n : n I} Hint: x2 – [x2] = {x2} > 0; but 0 {y} < 1 {x2} 0 x ± n ] Q.58192 If f(x) = cos x, x =n , n = 0, 1, 2, 3, ..... = 3, otherwise and (x) = x when x x when x when x 2 1 3 0 3 0 5 3 , then Limit x 0 f((x)) = (A) 1 (B*) 3 (C) 5 (D) none Q.59195 Let 0 x Lim sec–1 x x sin = l and 0 x Lim sec–1 x x tan = m, then (A*) l exists but m does not (B) m exists but l does not (C) l and m both exist (D) neitherl norm exists Q.60200 Range of the function f (x) = 2 2 x 1 1 ) e x ( n 1 l is , where [*] denotes the greatest integer function and e = / 1 0 ) 1 ( Limit (A) e 1 e , 0 {2} (B) (0, 1) (C) (0, 1] {2} (D*) (0, 1) {2} [Sol. ) e x ( n 1 2 l – 0 x 1 0 x 0 0 x x 1 1 0 x 2 ) x ( f 2 Hence range of f (x) is (0, 1) {2}] Q.61201 ] x [tan sin Lim 1 0 x = l then { l } is equal to (A) 0 (B) 2 1 (C) 1 2 (D*) 2 2 where [ ] and { } denotes greatest integer andfractional part function. [Hint: 2 ] x [tan sin Lim 1 0 x ; hence 2 = 2 2 2 = 2 – 2 Ans.]
- Q.62206 Number of points where the function f (x) = (x2 – 1) | x2 – x – 2 | + sin( | x | ) is not differentiable, is (A) 0 (B) 1 (C*) 2 (D) 3 [Hint: not derivable at x = 0 and 2 ] Q.63219 x 1 1 x 1 x 1 x 2 sec x 1 x cot Limit is equal to (A*) 1 (B) 0 (C) /2 (D)nonexistent [Hint: x 1 x Limit x = 0 cot–1(0) = /2 x x 1 x 1 x 2 Limit sec–1 () = /2 l = 1 Ans ] Q.64223 If f (x) = 0 0 2 x x if b ax x x if x derivable R x then the values of a and b are respectively (A*) 2x0 , – 2 0 x (B) – x0 , 2 2 0 x (C) – 2x0 , – 2 0 x (D) 2 2 0 x , – x0 Q.65231 Let f(x) = 1 2 1 1 2 1 2 2 1 4 2 1 2 1 2 cos sin , , , x x x p x x x x . If f(x) is discontinuous at x = 1 2 , then (A*) p R {4} (B) p R 1 4 (C) p R0 (D) p R [Hint: f 1 2 = 4 = f 1 2 ] Q.66235 Let f(x) beadifferentiable functionwhichsatisfies theequation f(xy) = f(x) + f(y) for all x > 0, y> 0 then f (x) is equal to (A*) f x '( ) 1 (B) 1 x (C) f (1) (D)f (1).(lnx) [Sol. f (x) = Limit h f x h f x h 0 ( ) ( ) = Limit h f x h x f x h 0 1 . ( ) = Limit h f x f h x f x h 0 1 ( ) = Limit h f h x x h x 0 1 = 1 0 1 1 x Limit t f t f t ( ) (note that f(1) = 0) = f x '( ) 1 Ans. ]
- Q.67240 Given f(x) = b ([x]2 + [x]) + 1 for x 1 = Sin ((x+a) ) for x < 1 where [x] denotes the integral part of x, then for what values of a, b the function is continuous at x = 1? (A*) a = 2n + (3/2) ; b R ; n I (B) a = 4n + 2 ; b R ; n I (C) a = 4n + (3/2) ; b R+ ; n I (D) a = 4n + 1 ; b R+ ; n I [Hint: f (–1) = b (1 – 1) + 1 = 1 ) h 1 ( Lim 0 h = 1 a ) h 1 ( Lim ) h 1 ( Lim 0 h 0 h forcontinuous sin a = – 1 = sin 2 3 n 2 a = 2 3 n 2 a = 2 3 n 2 hence a = 2 3 n 2 , n I and b R] Q.68247 Let f(x) = ) e x ( n ) e x ( n x 2 4 x 2 l l . If Limit x f(x) = l and Limit x f(x) = m then : (A*) l = m (B) l = 2m (C) 2 l = m (D) l + m = 0 [Sol. x Lim n x e n x e x x 2 4 2 = x Lim x 2 4 x 2 x 2 x e x 1 e n e x 1 e n l l = x Lim x 2 4 x 2 e x 1 n x 2 e x 1 n x l l = x / 1 x 2 4 x / 1 x 2 e x 1 n 2 e x 1 n 1 l l note that x as 0 e x x 2 and x as 0 e x x 2 2 = x Lim 1 e x 1 x 1 2 1 e x 1 x 1 1 x 2 4 x 2 = x Lim x 2 3 x e x 2 e x 1 |||ly 2 1 Limit x ] Q.69248 n Lim cos n n2 when n is an integer : (A) is equal to 1 (B) is equal to 1 (C*) is equal to zero (D) does not exist [Hint: n 1 1 1 2 n / = n 1 1 2 1 2 1 2 1 1 2 1 2 n n ! ...... = n n 1 2 1 2 1 2 1 1 2 1 ! ...... as n ; 2 . 2 1 1 2 1 1 2 1 n n ! ....... = (2n + 1) 2
- Alternatively (1): Take A = (2n +1) 2 and B = 2 .... n 1 ! 2 1 1 2 1 now cos (A + B) = cosA cosB – sinA sinB = 0 – (1) n Limit sin ... n 1 ! 2 1 1 2 1 2 = 0 ] Alternatively (2) Best l = n n n cos Limit 2 n n n n cos Limit 2 n n n n ) n ( cos Limit 2 n = n 1 1 n n n cos Limit n = n 1 1 1 cos Limit n = cos 2 0 ] Q.70250 0 x Limit x sin 3 ) x (sin ) x .(tan 7 x ) x 2 cos 1 ( ) x tan x (sin 5 6 1 7 1 5 4 2 is equal to (A) 0 (B) 7 1 (C*) 3 1 (D) 1 [Hint: DivideNr. andDr. byx5 andevaluate limit] Q.71264 Range of the function , f (x) = cot1 log / ( ) 4 5 2 5 8 4 x x is : (A) (0 , ) (B*) 4 , (C) 0 4 , (D) 0 2 , Q.72266 Let Limit x 0 [ ] x x 2 2 = l & Limit x 0 [ ] x x 2 2 = m , where [ ] denotes greatest integer , then: (A) l exists but m does not (B*) m exists but l does not (C) l & m both exist (D) neither l nor m exists . [Hint: [ ] x x 2 2 = 0 0 1 1 1 0 2 if x x if x l does not exist . [ ] x x 2 2 = 0 0 1 0 1 0 if x if x m exists and is equal to zero. ] Q.73268 The value of Limit x 0 tan { } sin { } { } { } x x x x 1 1 where {x} denotes the fractional part function: (A) is 1 (B) is tan 1 (C) is sin 1 (D*) is non existent [Sol. Limit x 0 tan { } sin { } { } { } x x x x 1 1 = 0 x Limit f (x) = ) 1 h ( h sinh . ) 1 h tan( Limit 0 h = 1 tan 1 ) 1 tan( 1 1 sin ) 1 h 1 ( ) h 1 ( ) h 1 sin( ) 1 ) h 1 tan(( Limit 0 x = sin 1 Hence limit x 0 f (x) does not exist ]
- Q.74270 If f(x) = n x e x x2 2 tan is continuous at x = 0 , then f (0) must be equal to : (A) 0 (B) 1 (C) e2 (D*) 2 [Hint: f (0) = Limit x 0 n x e x x2 2 = Limit x 0 l n e x x x 2 1 2 = Limit x 0 1 x e x x2 2 1 = Limit x 0 ex x 2 1 2 = 2 ] Q.75272 x Lim x sin e ) x 2 sin x 2 ( x 2 sin x 2 2 is : (A) equal to zero (B) equal to 1 (C) equal to 1 (D*)non existent [Sol. Limit x x sin e x x 2 sin 2 x x 2 sin 2 x 2 as x l = Limit x x sin e . 2 2 = oscillatory between e 1 to 1 e 1 non existent ] Q.76275 The value of x ec bx ax 0 2 lim cos cos is (A) e b a 8 2 2 (B) e a b 8 2 2 (C*) e a b 2 2 2 (D) e b a 2 2 2 [Sol. l = ) 1 ax (cos bx 2 ec cos 0 x Limit e now – bx sin ax cos 1 Limit 2 0 x = – ax cos 1 1 . bx sin ax sin Limit 2 2 0 x = – 2 2 b 2 a l = 2 b 2 2 a e ] Select the correct alternative : (More than one are correct) Q.77503 c x Lim f(x) does not exist when : (A) f(x) = [[x]] [2x 1], c = 3 (B*) f(x) = [x] x, c = 1 (C*) f(x) = {x}2 {x}2, c = 0 (D) f(x) = tan (sgn ) sgn x x , c = 0 . where[x] denotes step up function& {x} fractional part function.
- Q.78505 Let f(x) = tan { } [ ] { } cot { } 2 2 2 1 0 0 0 x x x x x for x for x for x where [x] is the step up function and {x} is the fractional part function of x , then : (A*) Limit x 0 f(x) = 1 (B) Limit x 0 f (x) = 1 (C*) cot -1 Limit f x x 0 2 ( ) = 1 (D) f is continuous at x = 1 . Q.79506 If f(x) = x n x n x x x . (cos ) 1 2 0 0 0 then : (A*) f is continuous at x = 0 (B)fiscontinuous atx=0 but notdifferentiableatx=0 (C*) f is differentiable at x = 0 (D) f is not continuous at x = 0. Q.80508 Whichofthefollowingfunction(s)is/areTranscidental? (A*) f (x) = 5 sin x (B*) f (x) = 2 3 2 1 2 sin x x x (C) f (x) = x x 2 2 1 (D*) f (x) = (x2 + 3).2x [Hint: functions whicharenotalgebraicareknownastranscidental function] Q.81513 Which ofthefollowingfunction(s) is/areperiodic? (A*) f(x) = x [x] (B) g(x) = sin (1/x) , x 0 & g(0) = 0 (C) h(x) = x cosx (D*) w(x) = sin1 (sinx) Q.82515 Which offollowingpairs offunctionsareidentical : (A) f(x) = e n x sec1 & g(x) = sec1 x (B*) f(x) = tan(tan1 x) & g(x) = cot(cot1 x) (C*) f(x) = sgn(x)& g(x) = sgn(sgn(x)) (D*) f(x) = cot2 x.cos2 x & g(x)= cot2 x cos2 x Q.83516 Whichofthefollowingfunctionsarehomogeneous ? (A) x siny+ ysinx (B*) x ey/x + y ex/y (C*) x2 xy (D) arc sin xy Q.84521 If is small & positivenumber then which of the following is/are correct ? (A) sin = 1 (B) < sin < tan (C*) sin < < tan (D*) tan > sin Q.85522 Let f(x) = x x x x . cos 2 1 & g(x) = 2x sin n x 2 2 then : (A) Limit x 0 f(x) = ln 2 (B) Limit x g(x) = ln 4 (C*) Limit x 0 f(x) = ln 4 (D*) Limit x g(x) = ln 2
- Q.86524 Let f(x) = x x x 1 2 7 5 2 . Then : (A*) Limit x 1 f(x) = 1 3 (B*) Limit x 0 f(x) = 1 5 (C*) Limit x f(x) = 0 (D*) Limit x 5 2 / does not exist Q.87527 Whichofthefollowinglimitsvanish? (A*) Limit x x 1 4 sin 1 x (B*) Limit x /2 (1 sinx) . tanx (C) Limit x 2 3 5 2 2 x x x . sgn(x) (D*) Limit x 3 [ ] x x 2 2 9 9 where [ ] denotes greatest integer function Q.88528 If x is a real number in [0, 1] then the value of Limit m Limit n [1 + cos2m (n! x)] is given by (A) 1 or 2 according as x is rational or irrational (B*) 2 or 1 according as x is rational or irrational (C) 1 for all x (D*) 2 for all x . [Sol. n m Limit Limit 1 + cos2m(n ! n) Let ] 1 , 0 [ x is a rational then n!. x if n becomes integral cos2 (n ! x) = 1 m Limit ( cos2 n ! x )m = 1 l = 1 + 1 = 2 ifxisrational if x is irrational , then n ! x is not an integer cos2( n! x) is less than 1 cos2 ( ( n! x) )m 0 l = 1 + 0 = 1 the result ] Q.89529 If f(x) is a polynomialfunction satisfyingthe condition f(x) . f(1/x) = f(x) + f(1/x)and f(2) = 9 then : (A) 2 f(4) = 3 f(6) (B*) 14 f(1) = f(3) (C*) 9 f(3) = 2 f(5) (D) f(10) = f(11) [Hint: f (x) = 1 + xn or 1 xn n = 3 f(x) = x3 + 1 ] Q.90531 Which ofthe followingfunction(s) not defined at x = 0 has/have removablediscontinuityat x =0 ? (A) f(x) = 1 1 2 cotx (B*) f(x) = cos x | x sin | (C*) f(x) = x sin x (D*) f(x) = 1 n x Q.91535 The function f(x) = x x x x x 3 1 1 2 4 3 2 13 4 , , is : (A*) continuous at x = 1 (B*) diff. at x = 1 (C*) continuous at x = 3 (D) differentiableat x = 3
- [Sol. f (x) = 1 x if 4 13 2 x 3 4 x 3 x 1 if x 3 3 x if 3 x 2 f (1+) = h ) 1 ( f ) h 1 ( f Limit 0 h = 1 h 2 ) h 1 ( 3 Limit 0 h f (1–) = h 2 4 13 ) h 1 ( 2 3 4 ) h 1 ( Limit 2 0 h = h 4 5 ) h 1 ( 6 ) h 1 ( Limit 2 0 h = h 4 h 6 h 2 h Limit 2 0 h = – 1 f is continuous at x =1 ] Q.92539 If f(x) = cos x cos 2 1 x ; where [x] is the greatest integerr function of x, then f(x) is continuous at : (A) x = 0 (B*) x = 1 (C*) x = 2 (D) none of these [Hint: Not defined at x = 0 ; f(x) = cos 3 ; f(2) = 0 and both the limits exist B and C ] Q.93540 Identifythepair(s) of functions which are identical . (A*) y = tan (cos 1 x) ; y = 1 2 x x (B*) y = tan (cot 1 x) ; y = 1 x (C*) y = sin (arc tan x) ; y = x x 1 2 (D*) y = cos (arc tan x) ; y = sin (arc cot x) [ Ans. : (A) T domain [ 1, 0) (0, 1] (B) T (C) T (D) T , 1 1 2 x ] (A) y= tan(cos–1x) note that 0 < sin–1x < p but tan(/2) is not defined hence cos–1x 2 x 0 hence domain is [–1,1] – {0} and range is R. The graphis as shown.
- (B) y= tan(cot–1x) note that 0 < cot–1x < tan(/2) is not defined hence cos–1x 2 x 0 hence domain = R – {0} since cot–1x 0 , y 0 hence range is R – {0} The graphis as shown. (C) y=sin(tan–1x) – 2 < tan–1x < 2 y (–1, 1) since tan–1x is defined for all x R and sin is also defined for all x R domain is R y = 2 x 1 x The graphis as shown. (D) y= cos(tan–1x) tan–1 2 , 2 y (0, 1] DomainisR y = 2 x 1 1 Note that sin–1(cot–1x) also has the same graph The graphis as shown.] Q.94544 The function, f(x) = [x][x] where [x] denotes greatest integer function (A*)is continuousforallpositiveintegers (B*)is discontinuous forallnonpositiveintegers (C*)hasfinitenumberofelementsinits range (D*) is such that its graph does not lie above the xaxis . [Sol. [ | x | ] – | [x] | = 2 x 1 0 1 x 0 0 0 x 1 1 1 x 0 range is {0, –1} Thegraphis ] Q.95545 Let f (x + y) = f(x) + f(y) for all x , y R. Then : (A) f(x) must be continuous x R (B*) f(x) may be continuous x R (C) f(x) must be discontinuous x R (D*) f(x) maybe discontinuous x R
- [Hint: Limit h0 f (x + h) = Limit h0 f(x) + f(h) = f (x) + Limit h0 f (h) Hence if h 0 f (h) = 0 'f' is continuous otherwise discontinuous ] Q.96548 The function f(x) = 1 1 2 x (A*) has its domain –1 < x < 1. (B*) has finite one sided derivates at the point x = 0. (C)iscontinuous and differentiable atx = 0. (D*)is continuous but not differentiable atx = 0. [Hint: f (0+) = 1 2 ; f (0–) = – 1 2 ] Q.97553 Let f(x) be defined in [–2, 2] by f(x) = max (4 – x2, 1 + x2), –2 < x < 0 = min (4 – x2, 1 + x2), 0 < x < 2 Thef(x) (A)iscontinuousatallpoints (B*)has apoint ofdiscontinuity (C) isnot differentiable onlyat onepoint. (D*) is not differentiableat morethan onepoint [Hint: f(x) max. (4–x2, 1+x2), –2 < x < 0 min. (4 - x2, 1+x2), 0 < x < 2] Q.98555 Thefunctionf(x)=sgnx.sinx is (A*)discontinuousno where. (B*) an evenfunction (C*) aperiodic (D)differentiableforallx [Hint: f (x) = sin sin x x x x 0 0 ] Q.99556 The function f(x) = x n 1 xl (A*)is aconstant function (B) has a domain (0, 1) U (e, ) (C*) is such that limit x1 f(x) exist (D)is aperiodic [Hint: y = x logx e = e (constant) Aand C as limit x1 f(x) = e] Q.100557 Which pair(s)of function(s) is/are equal? (A*) f(x) = cos(2tan–1x) ; g(x) = 1 1 2 2 x x (B*) f(x) = 2 1 2 x x ; g(x) = sin(2cot–1x) (C*) f(x) = e n x (sgn cot ) 1 ; g(x) = e n x 1 (D) f(x) = a X , a > 0; g(x) = a x 1 , a > 0 where {x}and [x] denotes thefractional part &integral part functions.
- Fill in the blanks: Q.101801 Afunction f is defined as follows, f(x) = sinx if x c ax b if x c where c is a known quantity. If f is derivable at x = c, then the values of 'a' & 'b' are _____ &______ respectively . [ Ans. : cos c & sin c c cos c ] Q.102802Aweight hangs byaspring& is caused to vibrate by a sinusoidal force. Its displacement s(t)at time t is given by an equation of the form, s(t) = A c k 2 2 (sin kt sin ct) where A, c & k are positive constants with c k, then the limiting value of the displacement as c k is ______. [Ans. : At kt k cos 2 ] Q.103805 Limit x 4 (cos ) (sin ) cos x x x 2 4 where 0 < < 2 is ______ . [Ans. : cos4 ln cos sin4 ln sin ] [Hint: put cos 2 = cos4 sin4 ] Q.104809 Limit x 0 cos / 2 3 2 x x has the value equal to ______ . [ Ans. : e6 ] Q.105812 If f(x) = sin x, x n, n = 0, ±1, ±2, ±3,.... = 2, otherwise and g(x) = x² + 1, x 0, 2 = 4, x = 0 = 5, x = 2 then Limit x0 g [f(x)] is ______ [Ans. : 1 ] [ IIT’86,2 ]

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