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