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JEE (Main+Advanced), , Limit, , Exercise – I, Section (A) : Definition of limet, LHL & RHL,, Indeterminate forms, 1., , 2., , 3., , 4., ,  4 x, x   0, , f ( x)   1,, x  0 , then  im f ( x), x 0, 3x 2 , x   0, , , If, , equals, (a) 0, (c) 3, , (b) 1, (d) Does not exist, , x, 2, , The value of  im   , where {.} represents, x 2, fractional part function, is, (a) 0, (b) 1, (c) -1, (d) Limit does not exists, Which of the following limit exists ?, , 1, x, , (a)  im  , x 0, ,  , , x 0, , 6., , The, , 8., , 13., , x 1/ 4, , of, ,  im sgn[tan x], where, x , , represents greatest integer function, is, (a) 0, (b) 1, (c) -1, (d) Limit does not exists, , [.], ,  im sin 1 (sec x) is equal to, x 0, , (a), 7., , value, , , , (b) 1, , 2, , (c) 0, ,  im, x 0, , (a) 1, (c) 0, ,  im, x 0, , 14., , 15., , x, , function) is, , [.] denotes greatest integer, , (a) an indeterminate form, (b) equal to 1, (c) equal to 0, , x 1, , [] denotes greatest integer function), (a) 0, (b) 1, (c) -1, (d) does not exist, If [x] = greastest integer  x, then  im( 1)[ x ] is, equal to (a) 1, (c) 1, , Let f ( x) , ,  im, x 1, , x 2, , (b) -1, (d) limit doesn’t exist, , x , , then  im f ( x) =, x , sin x, , x 1, equals 2x  7x  5, , (a) 1/3, (c) 1/2, ,  im, x 1, , 7, 3, , (c) , 16., , 0, 0, 17., , 2, , (b) -1/3, (d) -1/2, , 1 ,  1,  im( x 2  9) , ,  equals x 0,  x 3 x3, , (a), , x, equals x  x2, ,  x  (where, ,  im 1  x  [ x  1]  [1  x] is equal to (where, , (a) 2, (c) 0, , (d) Limit not exists, (b) -1, (d) Does not exist, , (a) an indeterminate form, (b) equal to 1, (c) not an indeterminate form, (d) equal to 2, , (a) -1, (b) 1, (c) 0, (d) does not exist, Section (B) : Factorisation, Rationalisation, Use of, standard limits, use of substitution, , (b)  im  x , , (c)  im x sin1/ x (d) All of the above, x 0, , 11., , x  /2, , x 0, , 5., ,  1 , 2 , x , , Which of the following limits exists (a)  im x x, , x , , 10., , (d)  im  sin x , , (c)  im 21/ x, ,  im x 2  1  x is, , 9., , 12., , (b)  im , x 0, , (d) not an indeterminate form, , (b) 4, (d) Does not exist, , 7x  2  3, 5x  1  6x  2, 14, 3, , (b) , (d), , 7, 6, , 7, 3, , a  2 x  3x, equal to :, 3a  x  2 x, 2, 2, (a), (b), 3, 3, 2, (c) 2, (d), 3 3, L im, xa, , (3 x  4)( x  1), , x 1, 2 x2  x  3, , lim, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 1

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JEE (Main+Advanced), 1, 10, 1, (c), 8,  x3/ 2  8 , lim , , x4,  x4 , (a), , 18., , 3, 2, 2, (c), 3, (a), , 19., , 2, , The value of, equals, , (a), , 1, 2a a  b, sin 3 x, lim, x  0 tan 5 x, 5, (a), 3, 3, (c), 5, tan mx, lim, x  0 n (1  5 x ), m, (a), 5, m, (c) , 5, ( x)5/3  (a )5/3, lim, x a, xa, (c), , 22., , 23., , 1, 4a, , (a) a 2/3, (c), , 5 2/3, a, 3, , 24., , 25., (b) 3, (d), , (c) 0, , 21., , 1, 10, 1, (d), 8, (b), , 1, 3, , 2, 8, 1, (d), 2, , 26., , (b), , xb  a b, ( a  b), x2  a2, ,  im, x a, , 1, a a b, 1, (d), 4a a  b, , (b), , 5, 3, 3, (d) , 5, , 27., , 28., , 5, m, , (d) , , , , x tan x, equals, (e  1)3/2, , L im, , e x  e  x, equal to x, , x 0, , (a) 0, (c) 1/2, x0, , x, , (b) 1, (d) 2, , (a) , , (b)   , , 29., , 5, m, , 5 5/3, a, 3, 3, (d) a 2/3, 5, , 31., , 32., , (b), , 33., , , , , (c) 12 p (n 4)3, , (d) 27 p (n 4) 2, , 2x 1, , x  0 (1  x )1/2  1, (a) log e 2, (b) log e 4, lim, , (c) log e, ,  im, x 0, ,  im, x 0, ,  im, x 0, , (a) -6, (c) 2, , 2, , sin 4 x, , 1, , (a) 0, (c) 1/2, 30., , (d), , (4 x  1)3, is equal to,  im, x 0,  x   x2 , sin   n  1  , 3,  p , (a) 9 p (n 4), (b) 3 p(n4)3, , (a) 4, (c) 10, , (b) , , (b), ,  im, , (c)   , , 2  1  cos x,  im, is equal to, x 0, sin 2 x, , (a), , 20., , Limit, , 1  x , , (d) log e 8, , equals (b) 8, (d) 12, , x(1  1  x 2 ), , 1  x 2 (sin 1 x)3, , equals, , (b) 1, (d) ¼, , 1  sin x  1  sin x, , x, (b) -2, (d) 1, , e x  e  x, equals, x 0 sin  x  sin  x, (a) 0, (b)   ,  im, , (c) -1, , (d) 1, , (a) 1, (c) 3/2, , (b) 0, (d) , , sec 4 x  sec 2 x, The value of  im, is, x  0 sec3 x  sec x,  im, x 0, , n(1  tan x ), , x, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 2

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JEE (Main+Advanced), , 34., , 35., , 36., , 37., , (a) 0, (c) -1, , Limit, , (b) 1, (d) Limit doesn’t exists, , 41., , n(2  x)  n0.5,  is equal to, x, 1, 3, (a), (b), 2, 2, 1, (c) , (d) 1, 2, n(1  nx),  im, , x 1, nx 2, 1, 1, (a) , (b), 2, 2, (c) 1, (d) Limit doesn’t exists, e tan x  e x,  im, is equal to, x  0 tan x  x, 1, (a), (b) 1, 2, 1, (c) , (d) -1, 2, ( x  6)1/3  2,  im, , x 2, 2 x, 1, 1, (a), (b) , 12, 12, 1, (c), (d) does not exist, 2,  im, x 0, , 42., , 43., , 44., , , , 39., 40., , , , , , The value of  im, x , , (a) 1, (c) -1, The, , , , , ( x 2  1), is 2x  1, , (b) 0, (d) ½, value, , 45., , 46., , , , ,  1, 8, n3 ,  im , ,  .... ,  is x  1  n 4, 1  n4, 1  n4 , , , (a) 1, , (c) , , 1, 4, , (b) 0, , (d) , , 1, 2, , (b) -1, (d) , , 47., , 48., , of, , 49., , 50., , (n  2)! (n  3)!, , n , (n  4)!, (b) , 1, (d), 2, xc  x  , , (b) 2c, (d) c, , If n  N , then  im, (a) 0, ,  im  x, n  , (a) c / 2, (c) 1/ c,  im, n , , , , , , , , , , ( x  a )( x  b )  x , , ab, ab, (b), 2, 3, ab, ab, (c), (d), 2, 3, n 1, n, 2n, 5 3 2,  im n, , n  5  2 n  32 n  3, (a), , 3x 2  1  2 x 2  1, The value of im, is 4x  3, 1, 1, (a), 3  2 (b), 3 2, 4, 4, 1, (c), 3 2, (d), 3 2, 2, , , , (a) 0, (c) 1, , (c) 1, , Section (C) : Infinite limits, , 38., , 1  5  52  .....  5n1, , n , 1  25n, ,  im, , (a) 5, (c) 1, , (b) 3, (d) zero, , (a) 0, (c) -1, , (b) 1, (d) , , 1, x 4 sin    x 2, x,  im, , 3, n , 1 x, ,  im, x , , (a) -1, (c) 0, , x  sin x, , x  cos 2 x, , (b) 1, (d) does not exist, ,  ( x  1)10  ( x  2)10  ...  ( x  100)10 ,  im , , x , x10  1010, , , 2, 3, (a) 10, (b) 10, (c) , (d) 104, , Section (D) : Use of expansion, L-Hospital rule, , e x  e  x  2 cos x,  im, , x 0, x sin x, , (a) 1, (c) -1, ,  im, x 0, , (b) 2, (d) -2, , 1  sin x  cos x  log e (1  x), , x3, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 3

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JEE (Main+Advanced), , 51., , 52., , 53., , 54., , (a) 1/2, (c) 0, ,  im, x 1, , 56., , 57., , 58., , (b) -1/2, (d) 1, , 59., , (b) -1, (d) 1/2, , 60., , 1  log e x  x, , 1  2x  x2, , (a) 1, (c) -1/2, , , , 1, 1 ,   is 1/3, 2h ,  h(8  h), , The value of  im , h 0, , (a) 1/12, (c) -16/3, , (b) -4/3, (d) -1/48, , axe x  bn(1  x)  cxe  x,  im,  2, then, x 0, x 2 sin x, a bc , , 61., ,  im tan x log e sin x , , 62., , (a) 12, (c) 36, x  / 2, , (a) 0, , (b) 1, (d), , The value of  im sec, , 1, 2, , , , 63., , log e x is -, , 2x, (a)  / 2, (b) 2 / , (c)  / 2, (d) 2 / , 2, x  x  ....  x n  n,  im, , x 1, x 1, x 1, , (a) n, , 64., , (b) 0, , (b) 1, (d) -1, , 65., , 66., , (a  x) sin(a  x)  a sin a,  im, , x 0, x, (a) a 2 cos a  2a sin a, (b) a (cos a  2sin a ), 2, , 2, ,   , 00 ,  0 ,1 ,  im, , theorem, , x , , of, , x, nx, ,  im, , Sandwitch, x x , e, x, , (b) -1, (d) 3, , (a) a, , (b) 0, , x, a, lim   cot  , x 0 x, a, , , 1, (c), a, lim, , x , , , , (a) 0, , 5, 2,  im , x  , , , (a) 1, (c) 0, , (d) 1, , , , x 2  5x  6  x , (b) , , a x, 2, , (d) , , 2, , 67., form, 68., , 5, 2, ,  ax  1 , (b) 2, (d) ½, , a x, 2, , 2, ,  1  , , , L im(1  tan x)1/ x , x 0, , (a) 1, , (b) e, , (d), , 1 x x, L im ,  , x 0,  1 x , 1, , (a) 1, (c) e 2, , 2 , , L im 1  2  , x ,  x , x2, , (a) 1, (c) e 2, , , , l im 1  tan 2 x, x 0, , (a) e5, (c) e, , (c) a 2 (cos a  2sin a ), , (d) a 2 cos a  2a sin a, Section, (E), :, Limits, , (a) 1, (c) 0, , (c) e 2, , n2, n(n  1), (c), (d), 2, 2, x, a, a x,  im x,  1, then a equals xa x  a a, (a) 0, (c) e, , 3 ,  1, lim , , , x 1 1  x, 1  x3 , , , (c), , (b) 24, (d) -12, , (c) -1, 55., , Limit, , , , 1, e, , (b) e, (d) e 2, (b) e, (d) e 2, , 5/ x, , , , (b) e 2, (d) e3, ,  x 1 , L im ,  , x ,  2x 1 , 1, (a), (b) 1, 2, 1, (c) 0, (d) e, 2, 1/1 x, L im(1  log e x), x 1, , x2, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 4

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JEE (Main+Advanced), (a), 69., , 1, e, , (d) e 2, , (c) e, , , , then  im 1  ax 2  bx  c, ,  x2  2 x  1 ,  im  2,  , x ,  x  4x  2 , , x 0, , (a) 0, , 73., , 74., , 75., , 76., , a  , , Answer Key, , (d) 2, , The value of  im  cos ec, , 1, a, 11, d, 21, c, 31, d, 41, a, 51, c, 61, d, 71, b, , x 0, , (b) -1, (d) 1/e, , x, , 1/ in, , x, , is, , , , log  x  , 2, , The value of  im, is x, tan x, x, (a) 0, (c) -1, , x2, , x  e x, , lim, , 2, , (b) 1, (d) 2, , (a) 0, (c) -1, , (b) 1, (d) limit does not exists, , (a) 0, (c) -1, , (b) 1, (d) limit does not exists, , nx, , x  x 3, , lim, , x, 2, x, (d), 3, (b), , (c) 2x, , (b) 1, , 1, 2, , (a) 1, (c) e, , is equal to, , (b) 2, (d) e, , x, , of, , [1.2 x]  [2.3x ]  ...  [n.(n  1) x], (where [.], n , n3, ,  im, , x, , lim(sin x) , , (c), , 72., , (d) e, , value, , (a) x, , (b) In a (   ), , (c) e a (   ), , (a) 1, (c) e 2, , , , 1, x , , The, , denotes the greatest integer function) is, , If  and  be the roots of ax 2  bx  c  0,, (a) a (   ), , 71., , 77., , (b) 1, , x , , 70., , Limit, , 2, d, 12, d, 22, c, 32, c, 42, a, 52, d, 62, d, 72, d, , 3, d, 13, b, 23, c, 33, b, 43, a, 53, b, 63, b, 73, a, , 4, d, 14, c, 24, b, 34, a, 44, c, 54, a, 64, c, 74, a, , 5, d, 15, c, 25, c, 35, a, 45, d, 55, b, 65, c, 75, a, , 6, d, 16, d, 26, d, 36, b, 46, c, 56, d, 66, a, 76, a, , 7, d, 17, a, 27, b, 37, b, 47, b, 57, b, 67, c, 77, d, , 8, d, 18, b, 28, b, 38, a, 48, a, 58, a, 68, a, , 9, a, 19, b, 29, c, 39, d, 49, b, 59, b, 69, c, , 10, c, 20, d, 30, d, 40, c, 50, b, 60, b, 70, c, , x 2, , The value of  im .   (where [.] denotes the, x 0 5  x , greatest integer function) is, (a), , 2, 5, , (c) 0, , (b) , , (d) , , 2, 5, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 5