Class 11 Mathematics Limits And Derivatives MCQs Set 15

Question. \( f(x) = x^2, \phi(a) = b^2, f'(a) = n \phi'(a) \) and \( \Phi'(a) \ne 0 \) then \( \operatorname{Lim}_{x \rightarrow a} \frac{\sqrt{f(x)} - a}{\sqrt{\phi(x)} - b} = \)
(a) \( b/a \)
(b) \( nb/a \)
(c) \( a/b \)
(d) \( na/b \)
Answer: (b) \( nb/a \)

Question. \( \operatorname{Lim}_{h \rightarrow 0} \frac{(2+h) \cos(2+h) - 2 \cos 2}{h} = \)
(a) \( \cos 2 - 2 \sin 2 \)
(b) \( \cos 2 + 2 \sin 2 \)
(c) \( \sin 2 - 2 \cos 2 \)
(d) \( \sin 2 + 2 \cos 2 \)
Answer: (a) \( \cos 2 - 2 \sin 2 \)

Question. Let \( f(x) = 3x^{10} - 7x^8 + 5x^6 - 21x^3 + 3x^2 - 7 \), then the value of \( \operatorname{Lt}_{h \rightarrow 0} \frac{f(1-h) - f(1)}{h^3 + 3h} \) is
(a) -50/3
(b) 22/3
(c) 13
(d) 53/3
Answer: (d) 53/3

Question. \( \operatorname{Lt}_{x \rightarrow 0} \frac{\log(1+x) - x}{x^2} = \)
(a) 1/2
(b) -1/2
(c) 1/3
(d) -1/3
Answer: (b) -1/2

Question. \( \operatorname{Lim}_{x \rightarrow 0} x^2 \sin \frac{\pi}{x} = \)
(a) 1
(b) 0
(c) does not exist
(d) \( \infty \)
Answer: (b) 0

Question. \( \operatorname{Lim}_{x \rightarrow \infty} \frac{x^2 (2 + \sin^2 x)}{x + 100} = \)
(a) 0
(b) 1
(c) \( \infty \)
(d) does not exist
Answer: (c) \( \infty \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{x}{a} \left[ \frac{b}{x} \right] \) (\( a \ne 0 \)) [where [ ] denotes the G.I.F.) is equal to
(a) a
(b) b
(c) \( b/a \)
(d) \( 1 - b/a \)
Answer: (c) \( b/a \)

Question. \( \lim_{x \to 0^+} (\sin x)^{\tan x} = \)
(a) \( e \)
(b) \( e^2 \)
(c) -1
(d) 1
Answer: (d) 1

Question. \( \lim_{n \to \infty} (\pi n)^{2/n} = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. \( \lim_{x \to \infty} \left( \frac{a^{1/x} + b^{1/x} + c^{1/x}}{3} \right)^x = \) (where a,b,c are real and non-zero)
(a) 0
(b) \( (abc)^{1/3} \)
(c) \( (abc)^{-1/3} \)
(d) 1
Answer: (b) \( (abc)^{1/3} \)

Question. \( \lim_{n \to \infty} \left( 1 + \sin\left(\frac{a}{n}\right) \right)^n = \)
(a) \( e \)
(b) \( e^a \)
(c) \( a^e \)
(d) \( a \)
Answer: (b) \( e^a \)

Question. If \( f(9) = 9, f'(9) = 4, \lim_{x \to 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3} = \)
(a) 4
(b) 1/4
(c) 1/2
(d) -1/2
Answer: (a) 4

Question. If \( f(a) = 2, f'(a) = 1, g(a) = -1, g'(a) = 2, \) then \( \lim_{x \to a} \frac{g(x)f(a) - g(a)f(x)}{x-a} = \)
(a) 1/5
(b) 5
(c) -1/5
(d) -5
Answer: (b) 5

Question. If \( f(x) = x \tan^{-1} x \) then \( \lim_{x \to 1} \frac{f(x) - f(1)}{x-1} = \)
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi+1}{4} \)
(c) \( \frac{\pi+2}{4} \)
(d) \( \frac{\pi+3}{4} \)
Answer: (c) \( \frac{\pi+2}{4} \)

Question. \( \lim_{x \to 0} \frac{1 - \cos x \cos 2x \cos 3x}{\sin^2 2x} = \)
(a) 3/2
(b) 5/2
(c) 7/4
(d) 9/2
Answer: (c) 7/4

Question. \( \lim_{x \to 0} x^3 \cos \left( \frac{2}{x} \right) = \)
(a) 0
(b) 1
(c) \( \infty \)
(d) does not exist
Answer: (a) 0

Question. \( \lim_{n \to \infty} \left( \frac{n}{n^2+1} + \frac{n}{n^2+2} + \frac{n}{n^2+3} + ... + \frac{n}{n^2+n} \right) = \)
(a) 0
(b) 1
(c) \( \infty \)
(d) does not exist
Answer: (b) 1

Question. \( \lim_{x \to -2^+} \frac{a e^{\frac{1}{|x+2|}} - 1}{2 - e^{\frac{1}{|x+2|}}} = \lim_{x \to -2^-} \sin \left( \frac{x^4-16}{x^5+32} \right) \) then a is
(a) \( \sin(3/5) \)
(b) 2
(c) \( \sin(2/5) \)
(d) \( \sin(1/5) \)
Answer: (c) \( \sin(2/5) \)

Question. If {x} denotes fractional part of x then \( \lim_{x \to 1} \frac{x \sin \{x\}}{x-1} \)
(a) 0
(b) -1
(c) 1
(d) does not exist
Answer: (d) does not exist

Question. \( \lim_{n \to \infty} \sum_{x=1}^{20} \cos^{2n}(x-10) = \)
(a) 0
(b) 1
(c) 19
(d) 20
Answer: (b) 1

Question. If \( f(x) \) is a real number in [0, 1], then the value of the function \( f(x) = \lim_{m \to \infty} \lim_{n \to \infty} (1 + \cos^{2m}(n! \pi x)) \) is given by
(a) 2 or 1 according as x is a rational or irrational
(b) 2 or 1 according as x is irrational or rational
(c) 1 for all x
(d) 2 for all x
Answer: (a) 2 or 1 according as x is a rational or irrational

Question. \( \lim_{x \to \infty} \sec^{-1} \left( \frac{x}{x+1} \right) \) is equal to
(a) 0
(b) \( \pi \)
(c) \( \pi / 2 \)
(d) Does not exist
Answer: (d) Does not exist

Question. If \( n \in N \) then \( \lim_{x \to n} (-1)^{[x]} \), where [x] denotes the greatest integer less than or equal to x, is equal to
(a) \( (-1)^n \)
(b) \( (-1)^{n-1} \)
(c) 0
(d) does not exist
Answer: (d) does not exist

Question. \( \lim_{x \to -1} \frac{\cos 2 - \cos 2x}{x^2 - |x|} \) is equal to
(a) 2
(b) sin 2
(c) 2 sin 2
(d) 0
Answer: (c) 2 sin 2

Question. The value of \( \lim_{x \to \infty} \left( 1 + \frac{1}{x^n} \right)^x, n > 0 \) is
(a) 1, if n < 1
(b) 1, if n > 1
(c) e, if n > 1
(d) e, if n < 1
Answer: (b) 1, if n > 1

Question. \( \lim_{x \to 0} \left[ 1 + x \left( 1 + \frac{f(x)}{kx^2} \right) \right]^{1/x} = e^3 \) and \( f(4) = 64 \) then K has value
(a) 1
(b) 2
(c) 4
(d) 5
Answer: (b) 2

Question. I : \( \lim_{n \to \infty} \frac{1+3+6+ \dots + \frac{n(n+1)}{2}}{n^3} = \frac{1}{6} \)
II : \( \lim_{n \to \infty} \frac{1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \dots + n \cdot n!}{(n+1)!} = 1 \)
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (c) Both I and II are true

Question. I : \( \lim_{\theta \to 0} \left( \left[ \frac{n \sin \theta}{\theta} \right] + \left[ \frac{n \tan \theta}{\theta} \right] \right) = \text{even integer} \) where [.] denotes the greatest integer function.
II : \( \lim_{x \to 0} \frac{ae^x - b}{x} = 2 \) then \( a = 2, b = 2 \)
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (b) Only II is true

Question. Arrange the following limits in the descending order
a) \( \lim_{x \to 3^-} \frac{3-x}{|x-3|} \)
b) \( \lim_{x \to 0} \frac{(1+x)^{1/8} - (1-x)^{1/8}}{x} \)
c) \( \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} \)
d) \( \lim_{x \to \infty} \{ \sqrt{x^2 + 6x + 7} + x \} \)
(a) b, c, d, a
(b) d, a, c, b
(c) d, c, b, a
(d) b, c, a, d
Answer: (b) d, a, c, b

Question. Assertion(A) : If \( |x| < 1 \) then \( \lim_{n \to \infty} (1+x)(1+x^2)(1+x^4)\dots(1+x^{2^n}) = \frac{1}{1-x} \)
Reason (R): \( (1+x)(1+x^2)\dots(1+x^{2^n}) = \frac{1-x^{4n}}{1-x} \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not the correct explanation of A
(c) A is true but R is false
(d) R is true but A is false
Answer: (a) Both A and R are true and R is the correct explanation of A

Question. Assertion(A) : \( \lim_{x \to \infty} \left( \frac{x^2 + 5x + 3}{x^2 + x + 2} \right)^x = e^4 \)
Reason(R) : \( \lim_{x \to \infty} (1 + x)^{1/x} = e \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not the correct explanation of A
(c) A is true but R is false
(d) R is true but A is false
Answer: (c) A is true but R is false

Question. Then the correct match for List - I from List - II
List - I
a) \( \lim_{x \to 0} \left( \frac{\sin x}{x} \right)^{\frac{1}{x^2}} \)
b) \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} \)
c) \( \lim_{x \to 0} (\cos x)^{\frac{1}{\tan x}} \)
d) \( \lim_{x \to a} \left( 2 - \frac{x}{a} \right)^{\tan\left(\frac{\pi x}{2a}\right)} \)
List - II
1) \( e^{-1/6} \)
2) \( e^{1/3} \)
3) \( e^{2/\pi} \)
4) 1
(a) a-1, b-4, c-3, d-2
(b) a-1, b-2, c-3, d-4
(c) a-2, b-1, c-4, d-3
(d) a-1, b-2, c-4, d-3
Answer: (d) a-1, b-2, c-4, d-3

Question. If m, n are +ve integers and \( a_0, b_0 \neq 0 \) are non-zero real numbers \( s = \lim_{x \to \infty} \frac{a_0x^m + a_1x^{m-1} + \dots + a_{m-1}x + a_m}{b_0x^n + b_1x^{n-1} + \dots + b_{n-1}x + b_n} \) then
List - I
a) If \( m = n \)
b) If \( m < n \)
c) If \( m > n \)
List - II
1) \( s = \frac{a_0}{b_0} \)
2) \( s = \infty \)
3) \( s = 0 \)
(a) a-1, b-2, c-3
(b) a-1, b-3, c-2
(c) a-2, b-1, c-3
(d) a-1, b-3, c-3
Answer: (b) a-1, b-3, c-2

Question. Match the following :
List-I
A) \( \lim_{x \to \infty} \left( \frac{a^{1/x} + b^{1/x} + c^{1/x}}{3} \right)^x \)
B) \( \lim_{x \to 5} \frac{x^2 - 9x + 20}{x - [x]} \)
C) \( \lim_{x \to 0} \frac{1 - e^{1/x^2}}{1 + e^{1/x^2}} \)
D) \( \lim_{n \to \infty} \frac{2 \cdot 3^n + 5 \cdot 2^n}{4 \cdot 3^n - 7 \cdot 2^n} \)
List-II
1) 1/2
2) -1
3) does not exist
4) \( (abc)^{1/3} \)
(a) A-1, B-2, C-3, D-4
(b) A-4, B-3, C-3, D-1
(c) A-4, B-3, C-2, D-1
(d) A-4, B-3, C-1, D-2
Answer: (c) A-4, B-3, C-2, D-1

Question. I : \( \lim_{n \to \infty} \frac{n!}{(n+1)! - n!} = 0 \)
II : \( \lim_{x \to 0} \frac{\sqrt{x - \sin^2 x}}{\sqrt{x + \cos x}} = 1 \)
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (c) Both I and II are true

Question. I : If \( a > 0 \) then \( \lim_{x \to \infty} \frac{[ax+b]}{x} = a \), [Where [x] denotes greatest integral part of x].
II : \( \lim_{x \to \frac{\pi}{2}} [\sin x] = 0 \) [Where [x] denotes greatest integral part of x].
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (c) Both I and II are true

Question. I : \( \lim_{x \to 0} \left( \frac{\tan[e^2]x^2 - \tan[-e^2]x^2}{\sin^2 x} \right) = 15 \), [Where [x] denotes greatest integral part of x].
II : \( \lim_{x \to 0} \left( \frac{1^x + 2^x + 3^x + \dots + n^x}{n} \right)^{1/x} = (n!)^{1/n} \)
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (c) Both I and II are true