Check the below NCERT MCQ Questions for Class 11 Maths Chapter 13 Limits and Derivatives with Answers Pdf free download. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. We have provided Limits and Derivatives Class 11 Maths MCQs Questions with Answers to help students understand the concept very well.

## Limits and Derivatives Class 11 MCQs Questions with Answers

Limits Class 11 MCQ Question 1.
The value of the limit Limx→0 (cos x)cot2 x is
(a) 1
(b) e
(c) e1/2
(d) e-1/2

Hint:
Given, Limx→0 (cos x)cot² x
= Limx→0 (1 + cos x – 1)cot² x
= eLimx→0 (cos x – 1) × cot² x
= eLimx→0 (cos x – 1)/tan² x
= e-1/2

Limits And Derivatives Class 11 MCQ Question 2.
The value of limit Limx→0 {sin (a + x) – sin (a – x)}/x is
(a) 0
(b) 1
(c) 2 cos a
(d) 2 sin a

Hint:
Given, Limx→0 {sin (a + x) – sin (a – x)}/x
= Limx→0 {2 × cos a × sin x}/x
= 2 × cos a × Limx→0 sin x/x
= 2 cos a

MCQ On Limits Class 11 Question 3.
Limx→-1 [1 + x + x² + ……….+ x10] is
(a) 0
(b) 1
(c) -1
(d) 2

Hint:
Given, Limx→-1 [1 + x + x² + ……….+ x10]
= 1 + (-1) + (-1)² + ……….+ (-1)10
= 1 – 1 + 1 – ……. + 1
= 1

Limits MCQ Class 11 Question 4.
The value of Limx→01 (1/x) × sin-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2

Hint:
Given, Limx→0 (1/x) × sin-1 {2x/(1 + x²)
= Limx→0 (2× tan-1 x)/x
= 2 × 1
= 2

Limits And Derivatives Class 11 MCQ Questions Question 5.
Limx→0 log(1 – x) is equals to
(a) 0
(b) 1
(c) 1/2
(d) None of these

Hint:
We know that
log(1 – x) = -x – x²/2 – x³/3 – ……..
Now,
Limx→0 log(1 – x) = Limx→0 {-x – x²/2 – x³/3 – ……..}
⇒ Limx→0 log(1 – x) = Limx→0 {-x} – Limx→0 {x²/2} – Limx→0 {x³/3} – ……..
⇒ Limx→0 log(1 – x) = 0

MCQ On Limits And Derivatives Class 11 Question 6.
Limx→0 {(ax – bx)/ x} is equal to
(a) log a
(b) log b
(c) log (a/b)
(d) log (a×b)

Hint:
Given, Limx→0 {(ax – bx)/ x}
= Limx→0 {(ax – bx – 1 + 1)/ x}
= Limx→0 {(ax – 1) – (bx – 1)}/ x
= Limx→0 {(ax – 1)/x – (bx – 1)/x}
= Limx→0 (ax – 1)/x – Limx→0 (bx – 1)/x
= log a – log b
= log (a/b)

Class 11 Limits MCQ Question 7.
The value of limy→0 {(x + y) × sec (x + y) – x × sec x}/y is
(a) x × tan x × sec x
(b) x × tan x × sec x + x × sec x
(c) tan x × sec x + sec x
(d) x × tan x × sec x + sec x

Answer: (d) x × tan x × sec x + sec x
Hint:
Given, limy→0 {(x + y) × sec (x + y) – x×sec x}/y
= limy→0 {x sec (x + y) + y sec (x + y) – x×sec x}/y
= limy→0 [x{ sec (x + y) – sec x} + y sec (x + y)]/y
= limy→0 x{ sec (x + y) – sec x}/y + limy→0 {y sec (x + y)}/y
= limy→0 x{1/cos (x + y) – 1/cos x}/y + limy→0 {y sec (x + y)}/y
= limy→0 [{cos x – cos (x + y)} × x/{y×cos (x + y)×cos x}] + limy→0 {y sec (x + y)}/y
= limy→0 [{2sin (x + y/2) × sin(y/2)} × 2x/{2y×cos (x + y)×cos x}] + limy→0 {y sec (x + y)}/y
= limy→0 {sin (x + y/2) × limy→0 {sin(y/2)/(2y/2)} × limy→0 { x/{y×cos (x + y)×cos x}] + sec x
= sin x × 1 × x/cos² x + sec x
= x × tan x × sec x + sec x
So, limy→0 {(x + y) × sec (x + y) – x×sec x}/y = x × tan x × sec x + sec x

Class 11 Maths Chapter 13 MCQ Question 8.
Limy→∞ {(x + 6)/(x + 1)}(x+4) equals
(a) e
(b) e³
(c) e5
(d) e6

Hint:
Given, Limy→∞ {(x + 6)/(x + 1)}(x + 4)
= Limy→∞ {1 + 5/(x + 1)}(x + 4)
= eLimy→∞ 5(x + 4)/(x + 1)
= eLimy→∞ 5(1 + 4/x)/(1 + 1/x)
= e5(1 + 4/∞)/(1 + 1/∞)
= e5/(1 + 0)
= e5

Limits MCQs With Answers Question 9.
The derivative of [1+(1/x)] /[1-(1/x)] is
(a) 1/(x-1)²
(b) -1/(x-1)²
(c) 2/(x-1)²
(d) -2/(x-1)²

Hint:
Let y = [1+(1/x)] /[1-(1/x)]
then dy/dx = [{1-(1/x)}*(-1/x²)]/[{1+(1/x)}*(1/x²)]
= (1/x²) [(1/x) -1 – 1 – (1/x)]/[1-(1/x)]²
= [-2/x²]/[(x-1)/x]²
= -2/(x-1)²

Limits And Derivatives MCQ Question 10.
The expansion of log(1 – x) is
(a) x – x²/2 + x³/3 – ……..
(b) x + x²/2 + x³/3 + ……..
(c) -x + x²/2 – x³/3 + ……..
(d) -x – x²/2 – x³/3 – ……..

Answer: (d) -x – x²/2 – x³/3 – ……..
Hint:
log(1 – x) = -x – x²/2 – x³/3 – ……..

MCQs Of Limits Class 11 Question 11.
If f(x) = x × sin(1/x), x ≠ 0, then Limx→0 f(x) is
(a) 1
(b) 0
(c) -1
(d) does not exist

Hint:
Given, f(x) = x × sin(1/x)
Now, Limx→0 f(x) = Limx→0 x × sin(1/x)
⇒ Limx→0 f(x) = 0

Ch 13 Maths Class 11 MCQ Question 12.
The value of Limn→∞ {1² + 2² + 3² + …… + n²}/n³ is
(a) 0
(b) 1
(c) -1
(d) n

Hint:
Given, Limn→∞ {1² + 2² + 3² + …… + n²}/n³
= Limn→∞ [{n×(n + 1)×(2n + 1)}/6]/{n(n + 1)/2}²
= Limn→∞ [{n×n×n ×(1 + 1/n)×(2 + 1/n)}/6]/{n × n ×(1 + 1/n)/2}²
= Limn→∞ [{n³ ×(1 + 1/n)×(2 + 1/n)}/6]/{n² ×(1 + 1/n)/2}²
= Limn→∞ [{(1 + 1/n)×(2 + 1/n)}/6]/[n4 × {(1 + 1/n)/2}²]
⇒ Limn→∞ [{(1 + 1/n)×(2 + 1/n)}/6]/[n × {(1 + 1/n)/2}²]
= [{(1 + 1/∞)×(2 + 1/∞)}/6]/[∞×{(1 + 1/∞)/2}²
= [{(1 + 0)×(2 + 0)}/6]/∞ {since 1/∞ = 0}
= {(1 × 2)/6}/∞
= (2/6)/∞
= (1/3)/∞
= 0
So, Limn→∞ {1² + 2² + 3² + …… + n²}/n³ = 0

MCQ Of Limits And Derivatives Class 11 Question 13.
The value of Limn→∞ (sin x/x) is
(a) 0
(b) 1
(c) -1
(d) None of these

Hint:
Limn→∞ (sin x/x) = Limy→0 {y × sin (1/y)} = 0

Class 11 Maths Limits MCQ Question 14.
The value of Limx→0 ax is
(a) 0
(b) 1
(c) 1/2
(d) 3/2

Hint:
We know that
ax = 1 + x/1! × (log a) + x²/2! × (log a)² + x³/3! × (log a)³ + ………..
Now,
Limx→0 ax = Limx→0 {1 + x/1! × (log a) + x²/2! × (log a)² + x³/3! × (log a)³ + …}
⇒ Limx→0 ax = Limx→0 1 + Limx→0 {x/1! × (log a)} + Limx→0 {x² /2! × (log a)²}+ ………
⇒ Limx→0 ax = 1

MCQ Of Limits Class 11 Question 15.
Let f(x) = cos x, when x ≥ 0 and f(x) = x + k, when x < 0 Find the value of k given that Limx→0 f(x) exists.
(a) 0
(b) 1
(c) -1
(d) None of these

Hint:
Given, Limx→0 f(x) exists
⇒ Limx→0 – f(x) = Limx→0 + f(x)
⇒ Limx→0 (x + k) = Limx→0 cos x
⇒ k = cos 0
⇒ k = 1

MCQ On Limits Class 11 Pdf Question 16.
The value of Limx→0 (1/x) × sin-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2

Hint:
Given, Limx→0 (1/x) × sin-1 {2x/(1 + x²)
= Limx→0 (2 × tan-1 x)/x
= 2 × 1
= 2

Limit Class 11 MCQ Question 17.
Limx→0 sin (ax)/bx is
(a) 0
(b) 1
(c) a/b
(d) b/a

Hint:
Given, Limx→0 sin (ax)/bx
= Limx→0 [{sin (ax)/ax} × (ax/bx)]
⇒ (a/b) Limx→0 sin (ax)/ax
= a/b

Class 11 Maths Ch 13 MCQ Question 18.
The value of the limit Limx→0 {log(1 + ax)}/x is
(a) 0
(b) 1
(c) a
(d) 1/a

Hint:
Given, Limx→0 {log(1 + ax)}/x
= Limx→0 {ax – (ax)² /2 + (ax)³ /3 – (ax)4 /4 + …….}/x
= Limx→0 {ax – a² x² /2 + a³ x³ /3 – a4 x4 /4 + …….}/x
= Limx→0 {a – a² x /2 + a³ x² /3 – a4 x³ /4 + …….}
= a – 0
= a

MCQs On Limits Class 11 Question 19.
If f(x) = (x + 1)/x then df(x)/dx is
(a) 1/x
(b) -1/x
(c) -1/x²
(d) 1/x²

Hint:
Given, f(x) = (x + 1)/x
Now, df(x)/dx = d{(x + 1)/x}/dx
= {1 × x – (x + 1)×1}/x²
= (x – x – 1)/x²
= -1/x²

Class 11 Maths Chapter 13 MCQ With Answers Question 20.
Limx→0 (e – cos x)/x² is equals to
(a) 0
(b) 1
(c) 2/3
(d) 3/2