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Limits And Derivatives

Class 11th Mathematics NCERT Exemplar Solution
Exercise
  1. lim_ { x arrow3 } { x^{2} - 9 }/{x-3} Evaluate :
  2. lim_ { x arrow {1}/{2} } frac { 4x^{2} - 1 }/{2x-1} Evaluate :
  3. lim_ { x arrow0 } { root {x+h} - sqrt{x} }/{h} Evaluate :
  4. lim_ { x arrow0 } { (x+2)^ { frac {1}/{3} } - 2^ { frac {1}/{3} }…
  5. lim_ { x arrow1 } { (1+x)^{6} - 1 }/{ (1+x)^{2} - 1 } Evaluate :…
  6. lim_ { x arrowa } { (2+x)^ { frac {5}/{2} } - (a+2)^ { frac {5}/{2} }…
  7. lim_ { x arrow1 } { x^{4} - root {x} }/{ sqrt{x}-1 } Evaluate :…
  8. lim_ { x arrow2 } { x^{2} - 4 }/{ root {3x-2} - sqrt{x+2} } Evaluate…
  9. lim_ { x arrow root {2} } { x^{2} - 4 }/{ x^{2} + 3 sqrt{2x+8} }…
  10. lim_ { x arrow1 } { x^{7} - 2x^{5} + 1 }/{ x^{3} - 3x^{2} + 2 }…
  11. lim_ { x arrow0 } { root { 1+x^{3} } - sqrt { 1-x^{3} } }/{ x^{2} }…
  12. lim_ { x arrow3 } { x^{3} + 27 }/{ x^{5} + 243 } Evaluate :
  13. lim_ { x arrow {1}/{2} } frac {8x-3}/{2x-1} - frac { 4x^{2} + 1 }/{…
  14. Find ‘n’, if lim_ { x arrow2 } { x^{n} - 2^{n} }/{x-2} = 80 , n inn…
  15. lim_ { x arrowa } {sin3x}/{sin7x} Evaluate :
  16. lim_ { x arrow0 } {sin^{2}2x}/{sin^{2}4x} Evaluate :
  17. lim_ { x arrow0 } {1-cos2x}/{ x^{2} } Evaluate :
  18. lim_ { x arrow0 } {2sinx-sin2x}/{ x^{3} } Evaluate :
  19. lim_ { x arrow0 } {1-cosmx}/{1-cosnx} Evaluate :
  20. lim_ { x arrow { pi }/{3} } frac { root {1-cos6x} }/{ sqrt{2} frac…
  21. lim_ { x arrow { pi }/{4} } frac {sinx-cosx}/{ x - frac { pi }/{4}…
  22. lim_ { x arrow { pi }/{6} } frac { root {3} sinx-cosx }/{ x - frac…
  23. lim_ { x arrow0 } {sin2x+3x}/{2x+tan3x} Evaluate :
  24. lim_ { x arrowa } {sinx-sina}/{ root {x} - sqrt{a} } Evaluate :…
  25. lim_ { x arrow { pi }/{6} } frac {cot^{2}x-3}/{cosecx-2} Evaluate :…
  26. lim_ { x arrow0 } { root {2} - sqrt{1+cosx} }/{sin^{2}x} Evaluate :…
  27. lim_ { x arrow0 } {sinx-2sin3x+sin5x}/{x} Evaluate :
  28. If lim_ { x arrow1 } { x^{4} - 1 }/{x-1} = lim_ { x arrowk } frac {…
  29. { x^{4} + x^{3} + x^{2} + 1 }/{x} Differentiate each of the functions…
  30. ( x + {1}/{x} ) ^{3} Differentiate each of the functions w.r. to x…
  31. (3x + 5) (1 + tan x) Differentiate each of the functions w.r. to x in…
  32. (sec x – 1) (sec x + 1) Differentiate each of the functions w.r. to x…
  33. {3x+4}/{ 5x^{2} - 7x+9 } Differentiate each of the functions w.r. to x…
  34. { x^{5} - cosx }/{sinx} Differentiate each of the functions w.r. to x…
  35. { x^{2}cos frac { pi }/{4} }/{sinx} Differentiate each of the
  36. (ax2 + cotx) (p + q cosx) Differentiate each of the functions w.r. to…
  37. {a+bsinx}/{c+dcosx} Differentiate each of the functions w.r. to x in…
  38. (sin x + cosx)2 Differentiate each of the functions w.r. to x in
  39. (2x – 7)2 (3x + 5)3 Differentiate each of the functions w.r. to x in…
  40. x2 sinx + cos2x Differentiate each of the functions w.r. to x in
  41. sin3x cos3x Differentiate each of the functions w.r. to x in
  42. {1}/{ ax^{2} + bx+c } Differentiate each of the functions w.r. to x in…
  43. Differentiate using first principle cos (x2 + 1) Differentiate each of…
  44. Differentiate using first principle {ax+b}/{cx+d} Differentiate…
  45. Differentiate using first principle x^ { {2}/{3} } Differentiate…
  46. Differentiate using first principle x cos x. Differentiate each of the…
  47. lim_ { y arrow0 } { (x+y) sec (x+y) - xsecx }/{y} Evaluate each of…
  48. lim_ { x arrow0 } { ( sin ( alpha + beta ) x+sin ( alpha - beta )…
  49. lim_ { x arrow { pi }/{4} } frac {tan^{3}x-tanx}/{ cosx + frac { pi…
  50. lim_ { x arrow pi } { 1-sin frac {x}/{2} }/{ cos frac {x}/{2} cos…
  51. Show that lim_ { x arrow4 } {|x-4|}/{x-4} does not exists Evaluate…
  52. Let f (x) = {kcosx}/{ pi -2x } x not equal frac { pi }/{2} and if…
  53. Let f (x) = {ll} {x+2}& { x less than equal to -1 } { cx^{2} }
  54. lim_ { x arrow pi } {sinx}/{ x - pi } is Choose the correct answer…
  55. lim_ { x arrow0 } {x^{2}cosx}/{1-cosx} is Choose the correct answer…
  56. lim_ { x arrow0 } { (1+x)^{n} - 1 }/{x} is Choose the correct answer…
  57. lim_ { x arrow1 } { x^{m} - 1 }/{ x^{n} - 1 } is Choose the correct…
  58. lim_ { x arrow0 } {1-cos4theta }/{1-cos6theta} is Choose the
  59. lim_ { x arrow0 } {cosecx-cotx}/{x} is Choose the correct answer out…
  60. lim_ { x arrow0 } {sinx}/{ root {x+1} - sqrt{1-x} } is Choose the…
  61. lim_ { x arrow { pi }/{4} } frac {sec^{2}x-2}/{tanx-1} is Choose…
  62. lim_ { x arrow1 } { ( root {x}-1 ) (2x-3) }/{ 2x^{2} + x-3 } is…
  63. If f (x) = [ {cc} { {sin[x]}/{[x]} , } & { [x] not equal 0 } { 0 ,…
  64. lim_ { x arrow0 } { | sinx| }/{x} is Choose the correct answer out…
  65. Let f (x) = { x^{2} - 1 , 0the quadratic equation whose roots are…
  66. lim_ { x arrow0 } {tan2x-x}/{3x-sinx} is Choose the correct answer…
  67. Let f(x) = x – [x], inr , f^ { there eξ sts } {1}/{2} is Choose…
  68. If y = root {x} + {1}/{x} , frac {dy}/{dx} at x = 1 is Choose the…
  69. If f (x) = {x-4}/{ 2 root {x} } then f’(1) is Choose the correct…
  70. If y = { 1 + frac {1}/{ x^{2} } }/{ 1 - frac {1}/{ x^{2} } } , frac…
  71. If y = {sinx+cosx}/{sinx-cosx} , frac {dy}/{dx} at x = 0 is Choose…
  72. If y = { sin (x+9) }/{cosx} frac {dy}/{dx} at x = 0 is Choose the…
  73. If f (x) = 1+x + { x^{2} }/{2} + l. s + frac { x^{100} }/{100} then…
  74. If f (x) = { x^{n} - a^{n} }/{x-a} for some constant ‘a’, then…
  75. If f(x) = x100 + x99 + … x + 1, then f’(1) is equal to Choose the…
  76. If f(x) = 1 – x + x2 – x3 … –x99 + x100, then f’(1) is equal to Choose…
  77. If f (x) = {tanx}/{ x - pi } then lim_ { x arrow pi } f (x) =…
  78. lim_ { x arrow0 } sinmxcot {x}/{ root {3} } = 2 then m = ________…
  79. If _______ Fill in the blanks
  80. lim_ { x arrow3^{+} } {x}/{[x]} = ___________ Fill in the blanks…
Exercise
Question 1.

Evaluate :
Answer:

Given


⇒



⇒


⇒



Question 2.

Evaluate :
Answer:

Given







⇒



Question 3.

Evaluate :
Answer:

Given







Question 4.

Evaluate :
Answer:

Given


Now put x=x-2 → limits also change from 0 to 2




Now we know that








Question 5.

Evaluate :
Answer:

Given










Question 6.

Evaluate :
Answer:

Given




Now we know that







Question 7.

Evaluate :
Answer:

Given



⇒


⇒


⇒


⇒


⇒


⇒



Question 8.

Evaluate :
Answer:

Given


⇒


⇒


⇒


⇒


⇒



Question 9.

Evaluate :
Answer:

Given


⇒


⇒



Question 10.

Evaluate :
Answer:

This question can be easily solved using LH rule i.e L.Hospital’s rule which says:


⇒


Given


⇒


⇒


⇒


⇒


⇒
Question 11.

Evaluate :
Answer:

Given


⇒


⇒


⇒


⇒


⇒



Question 12.

Evaluate :
Answer:

Given


⇒


⇒



Question 13.

Evaluate :
Answer:

Given


⇒


⇒


⇒


⇒


⇒


⇒


⇒



Question 14.

Evaluate :

Find ‘n’, if


Answer:

Given


Now we know that


⇒ ==


⇒ n=5
Question 15.

Evaluate :
Answer:

Given


⇒



Question 16.

Evaluate :
Answer:

Given


⇒


⇒


Now as


⇒


⇒
Question 17.

Evaluate :
Answer:

Given


⇒


⇒


⇒


⇒
Question 18.

Evaluate :
Answer:

Given


⇒


⇒


Now cosx=1-2sin2 (x/2)


⇒


⇒ =


⇒


⇒


Now as


⇒


⇒



Question 19.

Evaluate :
Answer:

Given


Here


⇒


And similarly


⇒


⇒


⇒


⇒


⇒


Now as


⇒


⇒



Question 20.

Evaluate :
Answer:

Given


Here


⇒


⇒


Now


⇒


⇒


Now as


⇒


⇒



Question 21.

Evaluate :
Answer:

Given


Here


⇒


⇒


⇒


⇒


Now as


⇒


⇒



Question 22.

Evaluate :
Answer:

Given


Here


⇒


⇒


⇒


⇒


Now as


⇒


⇒ =2
Question 23.

Evaluate :
Answer:

Given


⇒


⇒


Now as


⇒


⇒=1
Question 24.

Evaluate :
Answer:

Given


⇒


⇒


Now as


⇒ =


⇒


Now as


⇒


⇒ =
Question 25.

Evaluate :
Answer:

Given


Now as


⇒


⇒


⇒


⇒



Question 26.

Evaluate :
Answer:

Given


⇒


⇒


Now


⇒


⇒


⇒



Question 27.

Evaluate :
Answer:

Given


⇒


⇒


Now as


⇒


⇒=0
Question 28.

Evaluate :

If


Answer:

Given,
Question 29.

Differentiate each of the functions w.r. to x in
Answer:

Let y=


⇒


⇒


⇒


⇒


⇒



Question 30.

Differentiate each of the functions w.r. to x in
Answer:

Let


⇒




= 3x2 – 3x-4 + 3 – 3x-2
Question 31.

Differentiate each of the functions w.r. to x in

(3x + 5) (1 + tan x)
Answer:

Given


Applying product rule of differentiation i.e


⇒


⇒


⇒


⇒



Question 32.

Differentiate each of the functions w.r. to x in

(sec x – 1) (sec x + 1)
Answer:

Given


⇒


⇒


⇒ Now applying the concept of chain rule


⇒


⇒



Question 33.

Differentiate each of the functions w.r. to x in
Answer:

Given


Applying division rule of differentiation i.e


⇒


⇒







⇒



Question 34.

Differentiate each of the functions w.r. to x in
Answer:

Given


Applying division rule of differentiation i.e


⇒








Question 35.

Differentiate each of the functions w.r. to x in
Answer:

Given


Applying division rule of differentiation i.e
Question 36.

Differentiate each of the functions w.r. to x in

(ax2 + cotx) (p + q cosx)
Answer:

Given


Applying product rule of differentiation i.e
Question 37.

Differentiate each of the functions w.r. to x in
Answer:

Given


Applying division rule of differentiation i.e
Question 38.

Differentiate each of the functions w.r. to x in

(sin x + cosx)2
Answer:

Given


Applying the concept of chain rule
Question 39.

Differentiate each of the functions w.r. to x in

(2x – 7)2 (3x + 5)3
Answer:

This question will involve the concept of both chain rule and product rule.


Given


Applying product rule of differentiation i.e
Question 40.

Differentiate each of the functions w.r. to x in

x2 sinx + cos2x
Answer:

This question will involve the concept of both chain rule and product rule.


Given


Applying product rule of differentiation i.e
Question 41.

Differentiate each of the functions w.r. to x in

sin3x cos3x
Answer:

This question will involve the concept of chain rule .


Given








Question 42.

Differentiate each of the functions w.r. to x in
Answer:

This question will involve the concept of chain rule .


Given






Question 43.

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle cos (x2 + 1)
Answer:

Let f(x) = Cos (x2 + 1) ------------(i)


f(x + ∆x) = Cos [(x + ∆x)2 + 1] -------(ii)


Subtracting eq. (i) from eq. (ii),


f(x + ∆x) - f(x) = Cos [(x + ∆x)2 + 1] - Cos (x2 + 1)


Dividing both sides by ∆x,





As per the definition of differentiations,












Taking limits,


= -2 Sin (x2 + 1).1.(x) = -2x Sin (x2 + 1)


As


Hence, the required answer is -2x Sin (x2 + 1).
Question 44.

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle


Answer:

Let ----- (i)


------- (ii)


Subtracting eq. (i) from eq. (ii),



Dividing both sides by ∆x and taking the limit,




Using differentiation,





Taking limits, we have,




Hence the answer is.
Question 45.

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle


Answer:

Let f(x) = x2/3 ------ (i)


f(x + ∆x) =(x + ∆x)2/3 ------ (ii)


Subtracting eq. (i) from eq. (ii),


f(x + ∆x) - f(x) = (x + ∆x)2/3 - x2/3


Dividing both sides by ∆x and taking the limit,





By differentiation,




Expanding by binomial theorem, and rejecting the higher powers of ∆x as ∆x → 0





Hence, the answer is .
Question 46.

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle x cos x.
Answer:

Let y = x Cosx ----- (i)


y + ∆y = (x + ∆x) Cos (x + ∆x) ----- (ii)


Subtracting eq. (i) from eq. (ii),


y + ∆y – y = (x + ∆x) Cos (x + ∆x) - x Cosx


∆y = xCos (x + ∆x) + ∆x Cos (x + ∆x) – x Cosx


Dividing both sides by ∆x and take the limits,








Taking the limits,


=x(-Sin x)+Cos x



= -x Sinx + Cos x


Hence the answer is -x Sinx + Cos x.
Question 47.

Evaluate each of the following limits
Answer:









Taking the limits,



= x Sec x tanx + Secx


= Secx (x tan x + 1)


Hence, the answer is Secx (x tan x + 1).
Question 48.

Evaluate each of the following limits
Answer:


















Hence, the answer is .
Question 49.

Evaluate each of the following limits
Answer:








{Taking limits}





Hence, the answer is -4.
Question 50.

Evaluate each of the following limits
Answer:










Hence, the answer is .
Question 51.

Evaluate each of the following limits

Show that does not exists
Answer:



{|x - 4| = -(x - 4) if x < 4}



{|x - 4| = (x - 4) if x > 4}


LHL ≠ RHL


Hence, the limit does not exist.
Question 52.

Let and if find the value of k.
Answer:


and if

















k = 6


Hence the answer is 6.
Question 53.

Evaluate each of the following limits

Let find ‘c’ if exists.
Answer:

f(x) = x + 2, x ≤ -1


f(x) = cx2, x > -1





=1




= c


As the limits exist,


LHL = RHL


c = 1


Hence the answer is 1.
Question 54.

Choose the correct answer out of 4 options given against each Question

is

A. 1

B. 2

C. –1

D. –2
Answer:



= -1





Hence, the answer is Option (C).
Question 55.

Choose the correct answer out of 4 options given against each Question

is

A. 2

B.

C.

D. 1
Answer:







= 2 Cos0 = 2× 1 = 2



Hence Option (A) is the correct answer.
Question 56.

Choose the correct answer out of 4 options given against each Question

is

A. n

B. 1

C. –n

D. 0
Answer:




= n(1)(n-1)


= n



Hence Option (A) is the correct answer.
Question 57.

Choose the correct answer out of 4 options given against each Question

is

A. 1

B.

C.

D.


Answer:






Hence Option (B) is the correct answer.
Question 58.

Choose the correct answer out of 4 options given against each Question

is

A.

B.

C.

D. –1
Answer:










Hence Option (A) is the correct answer.
Question 59.

Choose the correct answer out of 4 options given against each Question

is

A.

B. 1

C.

D. –1
Answer:





Sin 2x = 2 Sinx Cosx







Hence Option (C) is the correct answer.
Question 60.

Choose the correct answer out of 4 options given against each Question

is

A. 2

B. 0

C. 1

D. –1
Answer:






Taking limits,





Hence Option (C) is the correct answer.
Question 61.

Choose the correct answer out of 4 options given against each Question

is

A. 3

B. 1

C. 0

D.


Answer:







= 1 + 1


= 2


Hence Option (D) is the correct answer.
Question 62.

Choose the correct answer out of 4 options given against each Question

is

A.

B.

C. 1

D. None of these
Answer:







Taking limits,





Hence Option (B) is the correct answer.
Question 63.

Choose the correct answer out of 4 options given against each Question

If where [.] denotes the greatest integer function, then is equal to

A. 1

B. 0

C. –1

D. None of these
Answer:


f(x) = 0, [x] = 0





= -1





= 1


LHL ≠ RHL


So, the limit does n’t exist.


Hence Option (D) is the correct answer.
Question 64.

Choose the correct answer out of 4 options given against each Question

is

A. 1

B. –1

C. does not exist

D. None of these
Answer:



= -1




= 1


LHL ≠ RHL


So, the limit does n’t exist.


Hence Option (C) is the correct answer.
Question 65.

Choose the correct answer out of 4 options given against each Question

Let the quadratic equation whose roots are

A. x2 – 6x + 9 = 0

B. x2 – 7x + 8 = 0

C. x2 – 14x + 49 = 0

D. x2 – 10x + 21 = 0
Answer:

f(x) = x2 – 1, 0 < x < 2


f(x) = 2x + 3, 2 ≤ x < 3



=[(2 - h)2 - 1] = (4 + h2 - 4h - 1)


=(h2 - 4h + 3) = 3


And, f(x) = 2x + 3


= [2(2 + h) + 3]


= 7


the quadratic equation whose roots are 3 & 7 is x2 – (3 + 7)x + 3(7) = 0, i.e.., x2 – 10x + 21 = 0.


Hence Option (D) is the correct answer.
Question 66.

Choose the correct answer out of 4 options given against each Question

is

A. 2

B.

C.

D.


Answer:







2x → 0


Hence Option (B) is the correct answer.
Question 67.

Choose the correct answer out of 4 options given against each Question

Let f(x) = x – [x], is

A. 3/2

B. 1

C. 0

D. –1
Answer:

f(x) = x – [x]


Checking for differentiability of f(x) at ,






= 1






= 1


LHD = RHD



Hence Option (B) is the correct answer.
Question 68.

Choose the correct answer out of 4 options given against each Question

If at x = 1 is

A. 1

B.

C.

D. 0
Answer:





= 0


Hence Option (D) is the correct answer.
Question 69.

Choose the correct answer out of 4 options given against each Question

If then f’(1) is

A.

B.

C. 1
D. 0
Answer:





f’(x) at x = 1 =



Hence Option (A) is the correct answer.
Question 70.

Choose the correct answer out of 4 options given against each Question

If

A.

B.

C.

D.


Answer:







Hence Option (A) is the correct answer.
Question 71.

Choose the correct answer out of 4 options given against each Question

If at x = 0 is

A. –2

B. 0

C.

D. does not exist
Answer:








= -2


Hence Option (A) is the correct answer.
Question 72.

Choose the correct answer out of 4 options given against each Question

If at x = 0 is

A. cos 9

B. sin 9

C. 0

D. 1
Answer:








= Cos 9


Hence Option (A) is the correct answer.
Question 73.

Choose the correct answer out of 4 options given against each Question

If then f’(1) is equal to

A.

B. 100

C. does not exist

D. 0
Answer:



f’(1) = 1 + 1 + 1 + ……. + 1 (100 times)


= 100


Hence Option (A) is the correct answer.
Question 74.

Choose the correct answer out of 4 options given against each Question

If for some constant ‘a’, then f’(a) is

A. 1

B. 0

C. does not exist

D.


Answer:





It does not exist.


Hence Option (C) is the correct answer.
Question 75.

Choose the correct answer out of 4 options given against each Question

If f(x) = x100 + x99 + … x + 1, then f’(1) is equal to

A. 5050

B. 5049

C. 5051

D. 50051
Answer:

f(x)=x100 + x99 + ………….x + 1


f'(x) = 100x99 + 99x98 +…… + 1


f'(1) = 100(1)99 + 99(1)98 + …… + 1



[Using Arithmetic Progression, where d = -1, a = 100 & n = 100]


= 50 (200 – 99)


= 50 (101)


= 5050


Hence Option (A) is the correct answer.
Question 76.

Choose the correct answer out of 4 options given against each Question

If f(x) = 1 – x + x2 – x3 … –x99 + x100, then f’(1) is equal to

A. 150

B. –50

C. –150

D. 50
Answer:

f(x) = 1 – x + x2 – x3 + …….- x99 + x100


f’(x) = -1 + 2x – 3x2 + ……. – 99x98 + 100x99


f’(1) = -1 + 2(1) – 3(1)2 + ……. – 99(1)98 + 100(1)99


= (-1 – 3 – 5…… -99) + (2 + 4 + 6 +….. + 100)
=


[Using Arithmetic Progression, where d = -2 & 2, a = -1 & 2 & n = 50 respectively]


= 25 (-2 – 98) + 25 (4 + 98)


= 25 (-100) + 25 (102)


= 25 (-100 + 102)


= 25 (2)


= 50


Hence Option (D) is the correct answer.
Question 77.

Fill in the blanks

If then ________
Answer:




= 1


Hence, the answer is 1.
Question 78.

Fill in the blanks

then m = ________
Answer:








Rationalizing the denominator,




Hence, the answer is .
Question 79.

Fill in the blanks

If _______
Answer:






Hence, the answer is y.
Question 80.

Fill in the blanks

___________
Answer:


------(i)




By using equation (i)




= 1


Hence, the answer is 1.