CBSE Class 11 Mathematics Chapter 13 Limits and Derivatives Multiple Choice Questions with Answers. MCQ Class 11 Mathematics Limits and Derivatives with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 11 Mathematics Limits and Derivatives MCQs with Answers to know their preparation level.
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MCQ Class 11 Mathematics Limits and Derivatives with Answers - Set - 2
Question 1:
Limx→0 (ex² – cos x)/x² is equals to
(a) 0
(b) 1
(c) 2/3
(d) 3/2
Correct Answer – (D)
Given, Limx→0 (ex² – cos x)/x²
= Limx→0 (ex² – cos x -1 + 1)/x²
= Limx→0 {(ex² – 1)/x² + (1 – cos x)}/x²
= Limx→0 {(ex² – 1)/x² + Limx→0 (1 – cos x)}/x²
= 1 + 1/2
= (2 + 1)/2
= 3/2
Question 2 :
The value of the limit Limx→0 {log(1 + ax)}/x is
(a) 0
(b) 1
(c) a
(d) 1/a
Correct Answer – (C)
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
Question 3 :
The value of Limx→0 (1/x) × sin-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2
Correct Answer – (C)
Given, Limx→0 (1/x) × sin-1 {2x/(1 + x²)
= Limx→0 (2 × tan-1 x)/x
= 2 × 1
= 2
Question 4 :
The value of Limx→0 ax is
(a) 0
(b) 1
(c) 1/2
(d) 3/2
Correct Answer – (B)
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
Question 5 :
The value of Limn→∞ {1² + 2² + 3² + …… + n²}/n³ is
(a) 0
(b) 1
(c) -1
(d) n
Correct Answer – (A)
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 Class 11 Mathematics Limits and Derivatives with Answers
Question 6 :
If f(x) = (x + 1)/x then df(x)/dx is
(a) 1/x
(b) -1/x
(c) -1/x²
(d) 1/x²
Correct Answer – (C)
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
Question 7 :
Limx→0 sin (ax)/bx is
(a) 0
(b) 1
(c) a/b
(d) b/a
Correct Answer – (C)
Given, Limx→0 sin (ax)/bx
= Limx→0 [{sin (ax)/ax} × (ax/bx)]
⇒ (a/b) Limx→0 sin (ax)/ax
= a/b
Question 8 :
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
Correct Answer – (B)
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
Question 9 :
The value of Limn→∞ (sin x/x) is
(a) 0
(b) 1
(c) -1
(d) None of these
Correct Answer – (A)
Limn→∞ (sin x/x) = Limy→0 {y × sin (1/y)} = 0
Question 10 :
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
Correct Answer – (B)
Now, Limx→0 f(x) = Limx→0 x × sin(1/x)
⇒ Limx→0 f(x) = 0
- NCERT Solutions Class 11 Mathematics Limits and Derivatives with Answers : Exercise 13.1
- NCERT Solutions Class 11 Mathematics Limits and Derivatives with Answers : Exercise 13.2
- NCERT Solutions Class 11 Mathematics Limits and Derivatives with Answers : Exercise 13 Misc
- NCERT Solutions Class 11 Mathematics Textbook download
