Limit (Practice Question)
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Practice Question Question based on
LEVEL-1 Q.7
Existence of limit
Limit Which of the following limits does not exist(A) lim
Q.1
4x, x 0 If f(x) = 1, x 0 , then lim f(x) equalsx 0 3x 2 , x 0
(A) 0 (C) 3
Q.2
x 0
(B) 1 (D) Does not exist
x 1 1, 3 1 x 1 x , If f(x) = then1 x , 1 x 2 3 x 2 , x2
(A) f(x) = 1 x 2
Q.3
x 2
Q.8
x,
x0 x 0 then, lim f(x) x 0 x 2 , x 0
Q.9
Q.4
Q.5
Q.6
lim x – [x] equals -
x3/ 2
(A) 0 (C) 1/2 Q.10
(B) 1 (D) 3/2
Which of the following limits exists(A) lim x |x| (B) lim [x] x 1 / 4
(C) lim x sin 1/x
(D) Does not exist
(D) All the above
x 0
Q.11
1 lim sin equalsx 0 x
lim
x a
1 ( x a ) 2 n 1
(A) (C) 0
(A) 0
(B) 1
(C)
(D) Does not exist
1 lim x sin equalsx 0 x
Q.12
(A) 1
(B) 0
(C)
(D) None of these
(n N) equals(B) – (D) Does not exist
e1 / x e 1/ x If f(x) = e1 / x e 1/ x , x 0 then 0, x 0 lim f(x) equals-
x 0
(A) 1 (C) 3
Let f(x) = x (–1)[1/x], x 0 where [ ] represent greatest integer function then lim f(x) is x 0
(A) 2 (C) –1
(B) 1 (D) does not exist
x 0
(C)
x 0
(A) 0 (C) 2
lim sin x equals-
(B) 0
(D) lim {x –|x|}
If f (x) = 1,
x
(A) 1
x 0
x 0
(B) lim f(x) =1 (D) lim f(x) = 0
(B) lim {x + |x|}
(C) lim |x|
x 1
(C) lim f(x) = –1
|x| x
(B) 0 (D) Does not exist
Q.13
(B) 2 (D) Does not exist
If f is a odd function and lim f(x) exists then x 0
lim f(x) equals-
x 0
(A) 0 (C) –1
(B) 1 (D) None of these
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1
Q.14
If [x] = greatest integer x, then lim (–1)[x] is
Q.22
x 2
equal to (A) 1 (C) ±1 Question based on
Q.15
Q.23
n 2 n 1 equals1 3 5 ..... (2 n 1)
lim
(A) 1 (C) 3/4
(B) 4/3 (D)
The value of lim
x
Q.17
2x 3 4x 7 3x 3 5x 2 4
(A) 2/3
(B) –7/4
(C) –4/5
(D)
The value of lim
n
(A)
Q.24
x sin x
lim
x cos 2 x
1 ( 3– 2) 4
(B)
(A) 0
(B) 1
(C)
(D) None of these
2 n 1 ..... lim is equal to1 n 2 1 n 2 1 n 2
is-
(A) 1 (C) –1/2
Q.25
Q.26
(2x 3) (3x 4) = (4x 5) (5x 6)
lim
(A) 0 (C) 1/5
1 e1 / x
x 0
1 ( 3+ 2) 4
(D) None of these
xe1/ x
lim
Q.27
equals(B) 1
(C)
(D) None of these
lim
(n 2)!(n 3)! equals(n 4)!
(A) 0 (C) 1
(B) 1/10 (D) 3/10
(B) 2 (D) 1/2
(A) 0
n
x
equals-
n
3n 2 1 2 n 2 1 is4n 3
(C) ( 3 – 2 ) Q.18
(B) 5 (D) None of these
x
n
Q.16
(A) 4 (C) e
(B) –1 (D) None of these
x
lim (4n + 5n)1/n equals-
n
(B) (D) None of these
2 3 n 1 lim 2 2 2 ..... 2 equalsn n n n
n
Q.19
sin 5x equalsx
lim
x
(A) 5 (C) 0
Q.20
(B) 1/5 (D) 1
Q.28
(A) 0
(B) 1/100
(C)
(D) None of these
lim
1 5 5 ..... 5
n
1 25 n
The value of
(A) 1 (C) –1/4
Q.29
n 1
(B) 1/2 (D) 2n
1 8 n3 ..... lim n 1 n 4 1 n 4 1 n 4
1 3 n 1 ... 2 2 2 isThe value of lim n 25n 2 n 3
2
Q.21
(A) 0 (C) 2n
is
(B) 0 (D) None of these
1 1 1 1 lim 2 3 .... n equals3 3 3 3
n
equals-
(A) 0
(B) –1
(C) 1
(D)
(A) 1/2 (C) 1
(B) 1/3 (D) 0
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2
Question based on
Q.30
The value of lim
1 sin 3 x cos 2 x
x / 2
(A) –
3 2
4
x 1
x 1 2x 7 x 5
lim
x 1
1 x 1 / 3 1 x 2/ 3
(B) 1/2 (D) – 2/3
equals(B) 1 (D) –1/3
The value of lim
xb a b x2 a2
x a
(A)
Q.40
1 4a
(B) 1
(A) 1 (C) 0 Q.41
1 x 1 x
lim
1 x 2 1 x 2
equals-
Q.42
2 1 cos x
lim
equals-
sin 2 x
2 8
(B)
2
x2 4x
(A) 0 (C) 1/4
lim (a 2 x 2 ax 1) (a 2 x 2 1) x
(C)
Question based on
Q.43
Q.44 lim
x 0
1 (1 x )
x 2 4a 2 1 a 1
1 2 a
(D)
3 a
lim
1 4 a
e x e x equalssin x sin x
(A) 0
(B) –
(C) –1
(D) 1
lim
x cos x sin x
x 0
equals-
(B)
equals-
Expansion method
x 0
(B) 2 (D) 1/2 sin 4 x
x 2a x 2a
lim
(A)
equals(B) 3/2 (D) None of these
equals(A) 1 (C) 0
(D) None of these
x 2a
(B) 1/2 (D) Does not exist x 3
lim
x 3
Q.37
4a a b
Rationalisation method
(A) 1 (C) 0
Q.36
1
(B) –1 (D) None of these
x 0
(A)
x 0
Q.35
a a b
The value of lim x3/2 ( x 3 1 – x 3 1 ) is-
(C) 0 Q.34
1
(D)
2a a b
(a > b) is -
x
equals-
(A) 1/3 (C) 2/3 Question based on
Q.39
equals(B) –1/3 (D) – 1/2
1 5 x
(A) 0 (C) 1/3
(C)
2
(B) 8 (D) None of these
3 5 x
lim
x 4
(B) 108 (D) None of these
(A) 1/3 (C) 1/2 Q.33
x 81 The value of lim is x 3 x 3
lim
Q.38
(D) 0
(A) –27 (C) undefined Q.32
is-
3 2
(B)
(C) 1
Q.31
(A) 4 (C) 10
Factorisation method
(A) 1/3
x 2 cos x
(B) 0
equals(C) 3
(D) –3
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3
Q.45
1 sin x cos x log (1 x )
lim
x3
x 0
Q.46
equals-
(A) 1/2
(B) – 1/2
(C) 0
(D) None of these sin 1 x tan 1 x
lim
x
x 0
Q.47
Q.53
3
Q.54
equals-
(A) 1
(B) –1
(C) 1/2
(D) –3/2
(B) 2
(C) –1
(A) a
(B) logea
(C) 1
(D) None of these
Let f(x) =
lim
x 3
1 18 x 2
(D) –2
Q.55
(B) –1/9 (D) None of these
The value of lim
x a
Q.48
lim
x 0
x.2 x x is equal to 1 cos x
(A) log 2
(B) log 4
(C) 0
(D) None of these
a x xa xx aa
(A) 0 (C) e
Q.56 lim
x 0
x tan x x
(e 1)
3/ 2
The value of lim (A) 4/5 (C) 3/8
equals-
(A) 0
(B) 1
(C) 1/2
(D) 2
Q.57
lim
x 0 Question based on
Q.50
Q.58
lim x log x equals-
x 0
(A) e (C) 1 Q.51
lim
x a
Q.52
(B) 1/e (D) 0 xm am xn an
(A) m/n
(B) 0
m m–n (C) a n
n n–m (D) a m
lim tan x log sin x equals-
(B) 1 (D) None of these
(B) 25/6 (D) None of these
(B) 1
(C) 2/3
(D) 1/3
1 2/3 1 1 x (B) x–2/3 (C) x 1/3 (D) 3x–2/3 3 3 3
lim
x x 2 ..... x n n equalsx 1
(A) n (C)
(B) 0
n2 2
(D)
n (n 1) 2
The value of lim [x tan x– (/2) sec x] isx / 2
x / 2
(A) 0 (C) –1
is-
(A)
x 1
Q.60
(32 3x )1 / 5 2
(x h )1 / 3 x1/ 3 lim equalsh 0 h
equalsQ.59
(16 5x )1 / 4 2
(1 sin x )1/ 3 (1 sin x )1/ 3 equalsx
(A) 0
L’ Hospital rule
= – 1, then a equals-
(B) 1 (D) –1
x 0
Q.49
, then the value of
f ( x ) f (3) isx 3
(A) 0 (C) – 1/3
e x e x 2 cos x equalslim x 0 x sin x
(A) 1
lim n[a1/n–1] equals-
n
(A) –1 (C) 1
(B) 0 (D) None of these
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4
Q.61
The value of lim h 0
(A) 1/12
Q.62
x / 2
x 2
(A) 0
Q.63
h (8 h )
1 is2h
Q.68
2
1 2
(D) –
(B) /2
The value of lim sec
(C) 1
(A) –5
1 2
Q.70
(C) –2
(D) 2
g ( x ) f (a ) g (a ) f ( x ) equalsx a
lim
(C)
(B) 1
If f(a) = 3, f ' (a) = –2, g(a) = –1, g'(a) = 4, then x a
x cos 2 isThe value of lim x 1 1 x
(A) /2
1 tan x is equal tolim x / 4 1 2 sin x
(A) 0 Q.69
x 1
Q.65
equals-
(B) 1
(A) 0 Q.64
1/ 3
(B) –4/3 (C) –16/3 (D) –1/48
1 sin x
lim
1
lim
h 0
(B) 10
(C) –10
(D) 5
(a h ) 2 sin (a h ) a 2 sin a is equal to h
(A) a2 cos a + 2a sin a (B) a (cos a + 2 sin a) (D)
(C) a2 (cos a + 2 sin a) (D) None of these
log x is2x
(B) 2/ (C) – /2 (D) –2/
Q.71
(1 x) (1 x )
The value of lim
sin 1 x
x 0
(A) 0
The value of lim cos x log (tan x) is-
(B) 1
(C) –1
is-
(D)
x / 2
(A) 1 (C) 0
(B) –1 (D) None of these
Q.72
3 1 lim equalsx 1 1 x 1 x 3
(A) 0
Q.66
lim
x 1
1 log x x 1 2x x
2
equals-
(A) 1 (C) –1/2 Q.67
(B) –1 (D) 1/2
The value of lim
Q.73
(B) – 1
The value of lim
x
(A) 0 Question based on
Q.74
(A) (B)
cos x + log sin x x cos x x
The value of lim (A) 0
(C) e5
(D) e– 5
log (1 kx 2 ) is 1 cos x
(C) k
The value of lim
cot px iscot qx
(A) 0
(B) 1 x 2 tan1/ x
lim
8x 2 7 x 1 1
x –
(C) x cos x + log sin x sin x (D) cos x log x + x
is-
(B) 1 x 0
Q.76
5x
(D) 1/3
Some standard limit
x 0
Q.75
x5
(B) 1
h 0
sin(x h ) log(x h) sin x log x ish
(C) –2
(A) – (C)
2 2 1 2
(D) 2k
(C) q/p
(D) p/q
is equal to (B)
1 2 2
(D) Does not exist
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5
Q.77
x 0
Q.78
(A) 1
(B) 2
lim
1 cos x 2 equals1 cos x
x 0
(A)
(C) 0
The value of lim ax sin (b/ax) is (a >1) x
(A) b log a (C) b
(D) 1/2
Q.89
(B) 1/ 2
2
(C) 1 Q.79
Q.88
sin 2 x equalsx cos x
lim
lim
x0
d dx
The value of lim (y –2) cosec a (y –2) is(A) 0
Q.90
(B) 1
(C) a
1 cos x x2
dx is equal to-
(A) 1/2 (C) 0
(D) None of these y 2
(B) a log b (D) None of these
(D) 1/a
(B) –1/2 (D) 1 cos = k, then value of 8x 8x
If lim x sin x
k isQ.80
Q.81
The value of lim n[log (n+1) – log n] is-
(A) /4
(B) /3
(A) 1
(C) /2
(D) /8
n
lim
x0
(B) 0
(C) –1
(D) 2
(1 x )1/ x e equalsx
(A) e
(B) e/2
(C) –e
Q.91 (D) –e/2
sin 2x lim x0 x
equals(C)
(A) 1 (C) 2 Q.83
lim
(B) 0 (D) None of these
2 sin 2 3x
x0
(A) 9
x2
Question based on
(B) 18
(C) 6
(D) 1
x
Q.85
If lim
x0
(A) 1 Q.87
(B) e
(C) 1
1 2
(D)
8
1 29
1 , 0 , 00 Forms
(B) 0
(C) 2
Q.93
(C) 5
1/ x
equals-
(A) e
(B) e–1
(C) e2
(D) e–1/2
lim [1 + tanx]
cot x
equals -
x 0
(A) 1 (D) 1/2
tan kx = 3, then the value of k issin 5x
(B) 3
log (1 x ) lim x 0 x
(D)
1 2x lim sin–1 is equal to x0 x 1 x 2
(A) 1 Q.86
2
x 1 lim equalsx 2x 1
(A) 0
(B) 1/24
equalsQ.92
Q.84
x0
equals(A) 1/16
1 x
Q.82
1 x2 x2 x2 x2 1 cos cos cos cos 2 4 2 4 x 8
lim
(D) 15
(C) Q.94
e–1
(B) e (D) None of these
lim (1+ x)1/x equals-
x 0
(A) 1 (C) e
(B) 0 (D) 1/e
lim x(e1/x – 1) equals-
x
(A) 0
(B) 1
(C) –1
(D)
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6
1/ x
Q.95
Q.96
1 x lim x 0 1 x
equals-
(A) e
(B) e2
(C) 1/e
(D) 1/e2
lim (sec x) cot x equals-
x / 2
(A) e (C) 1 Q.97
(B) 1/e (D) None of these
The value of lim (cosec x)1/log x is x 0
(A) 1 (C) e Q.98
The value of lim (tan x) tan 2x isx / 4
(A) e
Q.99
(B) –1 (D) 1/e
(B) e–1
x If f(x) = 2x
(C) 0
(D) –1
2x
, then-
(A) lim f(x) = e–6
(B) lim f(x) = 2
(C) lim f(x) = e–3
(D) lim f(x) = e–4
x
x
x
x
x
Q.100
a lim 1 equalsx x
(A) ax
Q.101
(B) e
4 lim 1 x x 1
(A) e2
(C) a
(D) ea
(C) e4
(D) e3
x3
(B) e
=
Q.102 The value of lim x1/x is x
(A) 0
(B) 1
(C)
(D) None of these
Q. 103 The value of lim (x + ex)2/x is x
(A) 1
(B) 2
(C) e
(D) e2
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7
LEVEL- 2 Q.1
sin x, x n, n Z and otherwise 2,
If f(x) =
Q.8
x 2 1, x 0, 2 g(x)= 4, x 0 then lim g[f (x)] = x 0 5, x2
(A) 0 Q.2
(B) 1
(C) 2
where [x] denotes the greatest integer x, lim f(x) equals -
x 0
(A) 1 (C) – 1
(D) 5
If [x] denotes the greatest integer x, then lim
n
1 n3
{[12 x] + [22 x] + [33 x] + …. + [n2 x]
Q.9
(A) 1/8 (C) 1/2
(A) x/2
(B) x/3
(C) x/6
(B) 0
(C) 1/2 1 x
The value of lim
(cos 1 x ) 2
x 1
(A) 1/2
(B) 1
Q.10
lim {log
n
and y and g(2) = 5, then lim g(x) is x 3
is-
(C) 1/4
(D) 4
k n – 1 (n) logn (n + 1) ... log n k 1 (n )},
(A) –8
(B) 10
(C) 8
(D) None of these
1 x 5 tan 2 3 | x | 2 7 x lim is equal to x | x |3 7 | x | 8
1
(A) –
k N is (A) 0 (C) does not exist
Q.6
If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y) + g (xy) – 2 for all real x
(D) 2
Q.11 Q.5
(B) 1/4 (D) 1
(D) 0
1 1 1 ..... lim equalsn 2.3 3.4 n (n 1)
(A) 1
Q.4
(B) 0 (D) None of these
1 x x 2 1 equalssin 4 x
lim
x 0
equals -
Q.3
sin(1 [x ]) for [ x] 0 If f(x) = [x ] 0 for [ x ] 0
(B) k (D) None of these
log x 2 The value of lim is tan x x
(B) 0
(C)
Q.12
lim
x 2
(D) does not exist 2 x 2 3 x 6
( 2 ) x 21 x
(A) 0
(B) 1
equals(C) 8
(D)
2
(A) 0 (C) –1
Q.7
(B) 1 (D) None of these
3 sin 6 h cos 6 h is equal to lim 2 h 0 3h ( 3 cosh sin h )
(A) 2/3 (C) –2
3
Q.13
lim
n
4n (1) n 5n (1) n
equals-
(A) 0 (C) 4/5 Q.14
lim
x
(B) (D) Does not exist x
equals-
x x x
(B) 4/3
(A) 0
(B) 1
(D) –4/3
(C)
(D) None of these
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8
Q.15
sec 4x sec 2x issec 3x sec x
The value of lim
x 0
Q.16
(A) 1
(B) 0
(C) 3/2
(D)
Q.23
e
x
g(x )
nx
1
(A) 1/2 (C) 1/4 Q.24
is -
(A) 0 (C) g(x)
(B) f(x) (D) None of these Q.25
Q.17
x (1 1 x 2 )
lim
1 x 2 (sin 1 x ) 3
x 0
(A) 0 (C) 1/2
equals-
(A) 0 (C) 1/2
(D) None of these
cos (sin x ) cos x x4
(A)
1 5
(B)
1 6
(C)
1 4
equals-
(D)
Q.28 Q.20
lim
x 0
x
(B) 24
(C) e3
(D) e4
lim
sin x n (sin x ) m
(m < n) is equal to(B) 1 (D) m/n
2 cos x 1
lim
( x ) 2
equals(B) 1/3 (D) 1/8
lim (log 5 5x ) log x 5 equals x 1
x
equals-
| x | x 2
(A) 1 (C) 0
(A) 1 (C) –1
(B) –1 (D) Does not exist Q.29 10
Q.21
(A) e2
(A) 1/2 (C) 1/4
1 2
2
is -
x 5x 3 The value of lim 2 isx x x 2
x
x 0
tan 2 x 4 tan x 3
(C)
(B) – 1/2 (D) None of these
The value of lim
tan 2 x 2 tan x 3
(B) 2
(A) 0 (C) n/m
Q.27 Q.19
lim1
(A) 0
x 0
2 n 1 .... lim is equal to2 2 n 1 n 1 n 1 n 2
equals-
(B) –1/2 (D) –1/4
The value of
(B) 1 (D) 1/4 Q.26
Q.18
x3
x tan 3
nx
f (x )e
x sin x log (1 x) x
x 0
If x > 0 and g is a bounded function lim
lim
10
( x 1) (x 2) .... ( x 100) lim x x10 1010
10
lim
x 0
is
(B) e (D) None of these
a
x
a1/
x
a
x
a 1/
x
(A) 1 (C) 0
(a > 1, x > 0) is equal to (B) –1 (D) None of these
equal to(A) 102
(B) 103
(C)
(D) 104
Q.30
lim
x / 2
Q.22
lim
x 4 sin (1 / x ) x 2
x
3
1 | x |
equals-
(A) 0
(B) 1
(C) –1
(D)
cot x – cos x is equal to( 2x )3 1 16
(A) 1
(B)
(C) 16
(D) None of these
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9
Q.31
If lim
x 0
x n sin x n x sin n x
must be (A) 1 (C) 3
Q.32
x 1 / 2
(C)
Q.33
1 tan (sin 1 x )
1
equals1
(B) –
2
1 2
2
1 2
(D) –
If f '' (0) = 4, then the value of 2f ( x) 3f (2x ) f (4 x)
lim
x2
x 0
(A) 11 Q.34
(B) 2 (D) None of these
x cos (sin 1 x )
lim
(A)
is non-zero definite, then n
(B) 12
is-
(C) 2
(D) 0
lim (x + (x– [x])2) equals-
x 2
where [x] represent greatest integer function. (A) 0 (B) 1 (C) 2 (D) 3
Q.35
x 1 1 x lim x tan 1 tan x 2 x 2
x
equals(A) 1
Q.36
(B) –1
(C)
1 2
lim
e x e sin x equalsx sin x
x 0
(D) –
1 2
(A) 0
(B) 1
(C)
(D) None of these
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LEVEL- 3 Q.1
If [x] denotes the greatest integer less than or equal to x, then [x ] [2x ] [3x ] ..... [nx]
lim
n2
n
(A) x/2 Q.2
(B) x/3
The value of lim (2 x )
b = lim
equals (D) 0
x 2
Let a = minm {x2 + 2x + 3} x R and 0
(C) x tan
Q.7
is equal to -
(A)
2n 1 – 1 3 .2 n
(C)
4n 1 – 1 3 .2 n
x 1
–2/
(B) e1/ (D) e–1/
(A) e (C) e2/ Q.3
lim
e
x [ a ]
– {x} – 1 is equal to where [·] {x}2
represent G.I.F. (A) 0 (C) e–2 Q.4
Let f (x), lim
n
x
2n
1
then -
(A) f (x) = 1, for |x| > 1 (B) f (x) = –1 for |x| < 1 (C) f (x) is not defined for any value of x (D) f (x) = 1 for |x| = 1 Q.5
Q.8
If f (x) = h (x) = –
Q.9
x 2 x 12
Q.10
(D) None of these
1x 2 x 3 x ...... n x lim x 0 n
Q.11
Q.12
(C) – Q.6
If Ai =
2 x 3x1/ 3 4 x1/ 4 ..... nx1 / n x ( 2 x – 3)1 / 2 ( 2 x – 3)1/ 3 ..... ( 2 x – 3)1/ n (A) 1 (B)
(D) None of these
2
sin x ] where [ ] x represent greatest integer function is (A) 5 (B) 6 (C) 7 (D) None of these
lim [(minm (y2 – 4y + 11)) x 0
If f(x) is the integral of
a1 < a2 < a3 < ... an. Then lim (A1A2 …..An),
2 sin x – sin 2 x , x 0 x3
x 0
(A) 1 (C) 3/2
(D) 0 x ai , i = 1, 2,..., n and if | x ai |
is equal to -
then find lim f (x) -
(B) –1
2 7
1/ x
(B) (n !)1/n (D) n(n !)
x 3
(A) –2
lim
(C)
then
lim [f(x) + g(x) +h(x)] is-
2n 1 1 3 .2 n
(B)
(B) e (D) None of these
(A) (n !)n (C) n !
2 x 3 , g (x) = and x 3 x4 2(2x 1)
Q.13
(B) 1/2 (D) None of these
If f (x) is a continuous function from f : R R 100
and attains only irrational value’s then
f (r) r 1
x a m
1 m n (A) is equal to (–1)m (B) is equal to (–1)m+1 (C) is equal to (–1)m–1 (D) does not exist
is -
r 1
sin x x sin x The value of lim x 0 x
(A) e–1 (C) 1
(B) 1/2 (D) None of these
x 2n 1
r n –r
a b
sin x
If { } represent fractional part of x then { x}
n
1 – cos the value of 2
is equal to 200
(A) 100
(B)
f (r )
r 101 10
(C)
f (r)
(D) None of these
r 1
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11
1cos( x 1)
Q.14
x 3 2x 2 x 1 The value of lim 2 x 1 x 2 x 3
is -
Q.19
e1 / x – 1
Q.20
(a, 0) through which the graph passes then log e (1 6 (f (x )) =2 3f ( x )
where [ ] represent G.I.F. and { } represent fractional part of x (A) lim f(x) = 1
Statement-II : Since the graph passes through
(B) lim– f(x) = cot 1
(a, 0). Therefore f(a) = 0, when f(a) = 0 given
(C) tan–1 lim f (x ) = /4 x 0 (D) All of the above
limit is zero by zero form. So that it can be
lim
x 0
lim
x a
The value of lim
x 0
log (x 2) x 2 n cos x
n
x 2n 1
x sin x 100 99 sin x x
as n , x2n 0. Q.22
1 Statement -II : lim y sin = 1 y y
(B) a = 1, b = 2 (D) None of these
All questions are Assertion & Reason type questions. Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Answer these questions from the following four option. (A) Statement-I and Statement-II are true Statement-II is the correct explanation of Statement-I (B) Statement-I Statement-II are true but Statement-II is not the correct explanation of Statement-I. (C) Statement-I is true but Statement-II is false (D) Statement-I is false but Statement-II is true.
1 x
Statement -I : lim x sin = 1 x 0
Statement type Questions
= log(x + 2)
Statement-II : For –1 < x < 1,
If lim 2 – (ax b) = 2 then x x 1 (A) a = 1, b = 1 (C) a = 1, b = – 2
Statement-I : when | x | < 1, lim
(B) 0 (D) None of these
where [ ] represent greatest integer function (A) 199 (B) 198 (C) 0 (D) None of these x3 1
evaluate by using L’Hospital’s rule. Q. 21
sin [cos x ] is 1 cos [cos x]
(A) 1 (C) does not exist
Q.18
Statement-I : The graph of the function y = f(x) has a unique tangent at the point
x 0
Q.17
1
Statement-II : lim 1 / x does not exist. x 0 e 1
x 0
Q.16
e
represent greatest integer function) does not exist.
Given a real valued function f such that tan 2 {x} , x0 2 2 ( x – [ x ] ) f(x) = 1 , x0 {x} cot{x}, x 0
e1 / x – 1 Statement – I : lim [x] 1 / x (where [ ] x 0
(B) e1/2 (D) None of these
(A) e (C) 1 Q.15
( x 1) 2
Q.23
Statement -I : lim
x 0
1 cos 2x 2 exist's. x
Statement -II : lim f(x) exists if the left hand x a
limit is equal to right hand limit.
Q.24
Statement -I : Value of lim (sinx)tanx is 1. x / 2
lim f ( x ) g ( x )
Statement -II: lim (1 + f(x))g(x) is e xa
,
x a
If lim f(x) = 0 and lim g(x) = x a
x a
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Passage Based Questions
Column Matching Questions Match the entry in Column 1 with the entry in Column 2.
Passage :- Let m, n are non zero integers and lim
x 0
tan mx n sin x x3
= an integer.
Q.30
On the basis of above information, answer the following questionsQ.25
Which of the following statement is true – (A) m is should be an even but n is odd (B) both m & n should be odd (C) m is odd and n is even (D) both m & n are even integers
Q.26
lim f(x) is less than equal to, where
x 0
Column-I x
(A) f ( x )
e e x
(B) f ( x )
ex ex sin x
(Q) – 2
(C) f ( x )
e2x e4x x
(R) – 1
2m n 2 6
(C)
(D) (1 + sin x)cosec x
(B)
2m 3 n 6
Q.31
Q.29
(S) 2
lim f(x), where f(x) is as in column-I is-
x 0
Column-I
(D) None of these
2
(A) f(x) =
Q.28
(P) e
The value of limit in terms of m & n is – (A)
Q.27
Column-II 2x
Is m & n are related as – (A) m2 = n (B) m = n2 (C) m = n (D) None of these The value of limit for m = 2 is – (A) 3 (B) 2 16 n (C) (D) None of these 12 If lim
x 0
tan (mx) n sin x x3
(B) f(x) =
Column-II 2
2
tan[e ]x tan[e ]x sin 2 x
(P)
2 /8
[5 / 2 tan x tan 2 x ] [5 / 2] (Q) 15 tan x
where [x] is the greatest integer function x cos x – log(1 x ) (C) f(x) = x2 (D) f(x) =
2
2 1 cos x sin 2 x
(R) 0 (S) 1/2
= not an integer then
for m = n = 1, the value of limit is– 1 1 (A) (B) – 2 2 (C) 2 (D) None of these
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13
ANSWER KEY LEVEL-1 Q.No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Ans.
A
C
D
D
B
B
A
A
C
D
D
D
A
D
A
A
A
D
C
B
Q.No.
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Ans.
A
B
B
C
A
A
B
C
A
B
B
B
B
D
D
D
B
D
D
A
Q.No.
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Ans.
B
B
D
B
B
C
B
B
B
D
C
A
B
D
B
B
C
B
D
A
Q.No.
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
Ans.
D
C
D
B
C
C
D
D
B
A
B
B
A
D
C
A
C
A
D
A
Q.No.
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Ans.
D
C
B
A
C
D
B
C
A
D
C
D
B
C
B
C
D
B
D
D
Q.No. 101 102 103 Ans.
C
B
D
LEVEL-2 Q.No.
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Ans.
B
B
C
C
B
A
B
B
A
B
Q.No.
21
22 23
24
25 26
27
28
29
30 31 32 33 34 35 36
Ans.
A
C
B
D
C
B
B
B
B
A
A A
C B
C B
B D
C C
B
C
B
B
D
B
LEVEL- 3 Q.No.
1
2
3
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Ans.
A
C
D A,B C
4
D
C
A
B
C
Q.No.
21
22
23
24
25
26
27
28
29
Ans.
A
D
D
A
D
B
C
A
A
30. (A) P,R,S ; (B) P,S ; (C) P,Q,R,S ; (D) P
B
A
B
D
D
B
B
C
B
A
31. (A) Q ; (B) R ; (C) S ; (D) P
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