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CENGAGE/GTEWANIMATHSSOLUTIONS
CHAPTER LIMITS AND DERIVATIVES || DPP DAILYPRACTICEPROBLEMS
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1
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
b.b=-4c. d.
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2
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
are finite real numbers, then b. c.d.
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3
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is b. c. d.
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4
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If beadifferentiablefunctionatg satisfying ,thenthevalueof
( lim )x→0
= 4, then
aex + b cos x + ce−x
e2x − 2ex = 1
a = 2 c = 2 a + b + c = − 8
(
lim )x−→∞
x loge(∣∣∣ 1γ0 β10 ∣∣∣)
= − 5, whereα, β, γ
α
x
1x
1x
α = 2, β = 1, γ ∈ R α = 2, β = 2, γ = 5α ∈ R, β = 1, γ ∈ R α ∈ R, β = 1, γ = 5
( lim )x−→3π4
1 + tanx31 − 2 cos2 x
−1/2 −2/3 −3/2 −1/3
f :R→R x = 0 f(0) = 0andf ′ (0) = 1
1
b. c. d.
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5
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is b. c. d.1
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6
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
hasthevalueequalto b.0c. d.
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
( lim )x→0
∞
∑n=1
( − 1)nf( ) =
1x
x
n
e −log2 0 1
( lim )x→0
.e
( )11+2x
1+3x
x2
1x e
52 e2 e−2
( lim )x→0
+
(atan−1
− b )
1
x√x
√x
a
tan−1(√x)
b
a − b
3
(a2 − b2)
6a2b2
a2 − b2
3a2b2
7
If
b. c. d.noneofthese
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8
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof isa.1b.2c.4d.8
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9
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
b. c. d.
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10
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is b. c. d.
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11
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Numberofintegralvaluesof forwhich
doesnotexistisa.1b.2c. d.4
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f ′ (a) = ,
then(
lim )h→0
=
14
f(a + 2h2) − f(a − 2h2)
f(a + h3 − h2)
− f(a − h3 + h2)
0 1 −2
( lim )x→0
ex2 − ex + x
1 − cos2x
( lim )x→1
=1 + log x − x
1 − 2x + x21 −1 0 −1/2
( lim )x→1
x − x113
17
(x − x )15
13
4491
4591
4589
4093
λ
( lim )x→1sec−1(
− )
λ2
(log)ex
λ2
x − 13
12
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
and
Thenthevalueof a.is-2b.is-3c.is1d.doesnotexist
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13
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
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14
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
f(x) = { , x > 02
− x, x ≤ 0
x
s ∈ x
g(x) = {x + 3, x < 1x2
− 2x − 2, 1 ≤ x < 2x
− 5, x ≥ 2( lim )
x→0
g(f(x))
limx→ ∞
n
∑r=1=
(x + r)2010
(x1006 + 1)(2x1004 + 1)
( lim )x→0
− aand(
lim )x→0
= b(whereb ≠ 0).
f(x)
x2
f(1 − cosx)
g(x)sin2 x
(1 − cos 2 ) 4
then is b. c. d.noneofthese
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15
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is b. c. d.
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16
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
b. c. d.
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17
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
( lim )x→0
g(1 − cos 2x)
x4
4a
b
a
4b
a
b
( lim )x→π / 4
tan3 x − tanx
cos(x + )π
4
8 4 −8 −2
(lim
)x→π
2
(1 − s ∈ x)(8x3 − π3)cosx
(π − 2x)4
π2
63π2
16π2
16−3π2
16
limx → π
4
√1 − √sin 2x
π − 4x
18 Thevalueof is b. c. d.
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19
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If arepositiveintegersand
a.2b. c. d.
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20
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
,then (where[.]denotesthegreatestintegerfunction)
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21
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
(where[.]denotesthegreatestintegerfunction)a.is0b.is1c.is-1d.doesnotexist
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22
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Let
Which of the following is true? Both exists Neither existsexistsbut doesnotexist existsbut doesnotexist
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( lim )x−→∞
(e√x4+ 1 − ex2+1) 0 e 1/e −∞
anandbn
an + √2bn = (2 + √2)n,
then( lim )x−→∞
( ) =an
bn
√2 e√2 e2
f(x) = {x + , x < 02x
+ , x ≥ 0
12
34
[( lim )x→0f(x)] =
( lim )x−→− 7
=
[x]2 + 15[x] + 56
sin(x + 7)sin(x + 8)
L1 = ( lim )x→4(x
− 6)xndL2 = ( lim )
x→4(x
− 6)4.L1andL2 L1andL2 L1
L2 L2 L1
23
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If isdefinedby for thenisequalto(where[.]representsthegreatestintegerfunction)
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24
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If denotes thegreatest integer function, then b. c. d.
doesnotexist
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25
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
(where [.] denotes the greatest integer function) a. -9 b.
-12c.-6d.0
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
f :R→R f(x) = [x − 3] + |x − 4| x ∈ R, ( lim )
x→3
f(x)
[.] ( lim )x→0
[ ]x
a
b
x
b
a0
a
b
( lim )x
[ ] =−−−−→− 1−
3
1x
−1x
26
a.0b. c. d.
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27
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
isequaltoa.2b.6c.-4d.4
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28
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Ifgraphofthefunction iscontinuousandpassesthroughpoint then
isequal b. c. d.
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29
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Setofallvaluesof suchthat
is non-zero and finite number when is b. c.
d.
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(
lim
)x−→∞
x2 sin((log)e√ )cosπ
xπ2
2π2
4π2
8
( lim )x→0
∣∣1
− cos3x(log)e(1
+ 4x)sin−1(ex
− 1)tan−1(2x)∣∣
1x2
y = f(x) (3, 1)
( lim )x→3
(log)e(3f(x) − 2)
2(1 − f(x))32
12
−32
−12
x(lim
)x−→∞
1
1 + ( )4n4 tan− 1 ( 2πx )
π
n ∈ N (0, )12π
( − , )1π
1π
[ , ]12π
12π
( − , 0)12π
30
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
b.0c. d.
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31
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof (b) (c) (d)noneofthese
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32
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If thenthevalueof is b. c. d.
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33
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
( lim )x−→∞
[x
− loge( )] =ex + e−x
2(log)e4 ∞ (log)e2
(lim)x→0
is
1 + sinx − cos x
+ log(1 − x)
x312
−12
0
( lim )x−→∞
( )x
= 4x + c
x − cec 1/4 1/2 1 2
exist,thenThevalueof is b.1c. d.
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34
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
thenfindthevalueof
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35
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Let bethefourthdegreepolynomialsuchthat
Thevalueof is b.1c. d.
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f(x)
= (
lim
)x−→∞
and(
lim )x−→± 1
f(x)
(x2 + ax + 1)
+ x2n(2x2 + x + b)
1 + x2n
b −1 0 2
(lim)x→0
[1 + x + ]= e3,
f(x)x
1x
1n⎛⎝(lim)
x→0
[1 + ]⎞⎠is __
f(x)x
1x
f(x)f ′ (0) − 6, f(0)
= 2and( lim )x→1
= 1
f(x)
(x − 1)2
f(2) 3 0 2
36
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Let bethefourthdegreepolynomialsuchthat
Thevalueof is b.1c. d.
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37
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
isequalto b. c. d.
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38
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof is
f(x)f ′ (0) − 6, f(0)
= 2and( lim )x→1
= 1
f(x)
(x − 1)2
f(2) 3 0 2
(
lim )x→0
()
√1 + xs ∈ x − √cos 2x
tan2(x/2)
16
6 3 2
limx → ∞
(x2)(1 − cos( ))1x
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39
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
isequalto
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40
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
(where[.]denotesthefractionalpartfunction)a.0b.2c.1d.doesnotexist
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41
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof isa. b.2c. d.noneofthese
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
limx→ ∞
cot−1(√x + 1 − √x)
sec− 1{( )x)2x+1x−1
lim x
→ (2+ ){x} =sin(x − 2)
(x − 2)2
( lim )x → 0√1 − cos x2
1 − cos x
12
√2
42
exist,thenThevalueof is(a) (b).1(c.) (d).
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43
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
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44
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
=
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45
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof
isa.9b.18c.27d.1/3
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46
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If (finite), then b. c.
d.
f(x) =
limn→ ∞
and limx → ±1
f(x)
(x2 + ax + 1)
+ x2n(2x2 + x + b)
1 + x2n
b −1 0 2
limx→0+
⎛⎜⎜⎜⎝(1 −
)
cos(sin2 x)
x2
ln( 1−2x2)sin2x
limx→ ( )
(1 − sin x)tanxπ
2
(lim
)x→3
(x3 + 27)(log)e(x − 2)
x2 − 9
( lim )x→0
= beax − ex − x
x2a = 2, b = 0 a = 0, b =
32
a = 2, b =32
a = 0, b = 2
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47
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
isequaltoa.36b.37c.38d.40
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48
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
(where[.]denotesthegreatest integerfunction)a.0b.1
c.-1d.noneofthese
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
( lim )x→0
= 1, a > 0thena + b
x3
√a + x(bx − sinx)
( lim )x→0
[ ] =sin−1 x
tan−1 x
49Findthevalueof
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50
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalue/valuesofais
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51
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If then (where [.] represents the greatest integer
function) b. c.
d.
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52
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If and then (where [.] denotes greatest
integerfunction)(A)A=1(B)Adoesnotexist(C)B=0(D)B=1
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53
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If isaperiodicfunction,then
isequalto b. c. d.-1
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limx→
( )π
4
√1 − √sin 2x
π − 4x
L = limx →0
= − ( )
sin(sin x) − sinx
ax5 + bx3 + c
112
f(x) = x( )e | x | + [ x ] − 2
|x| + [x]
( lim )x→0
+ f(x) = − 1 ( lim )x→0
− f(x) = 0
( lim )x→0
□ f(x) = − 1 ( lim )x→0
□ f(x) = 0
A = limx→0
sin−1(sinx)
cos−1(cosx)B = lim
x→0
|x|
x
conx
sinax( lim )
m−→∞( lim )
n−→∞(1
+ cos2m n !πa)0 1 2
54
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
where{.}denotesthefractionalpartofxisequalto
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55
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Assumethat existsand
holdsforcertainintervalcontainingthepoint then isequal
to b.isequalto1c.isnon-existentd.isequalto-1
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56
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
Thevalueof
is b. c. d.
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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
limx→ ∞
{(ex + πx) } =1x
( lim )θ→1
f(θ)
≤
≤
θ2 + θ − 2θ + 3
f(θ)
θ2
θ2 + 2θ − 1θ + 3
θ = − 1 ( lim )θ−→− 1
f(θ)f( − 1)
( lim )n−→∞
()2
√n2+n − 1
√n2 + n − 1n
e 1/e e2 e−2
57Let be such that Then
isa. b. c. d.
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58
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
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59
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If then
b. c. d.
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60
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
a.b.c.d.doesnotexist
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f :R→R f(a) = 1, f ′ (a) = 2.
( lim )x→0
( )1/ xf 2(a + x)
f(a)e2 e4 e−4 1/e
limx→ ∞
(1 − x + x. e )n1n
a > 0, b > 0
( lim )n→ ∞ ( )n
=
a − 1 + b1n
a
b1 /a a1b ab ba
( lim )x
[1 + (cosx)cosx]2
=
→π
2
61
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
isequalto b. c. d.
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62
CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_LIMITS ANDDERIVATIVES_Miscellaneous
If
,thenthevalueof is b. c.2d.
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(lim
)x→0
log(ex ^ 2 + 2√x)
tan√x0 1 2 e2
f(x)
= ( lim )x−→∞
( )ncos x
√n
( lim )x→0
f(x) − 1
x0 1 3/2
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