Class 11 Mathematics Limits And Derivatives MCQs Set 18

Question. \( \lim_{x \to 0} \frac{e^{x^3} - 1 - x^3}{\sin^6(2x)} = \)
(a) \( \frac{1}{128} \)
(b) \( \frac{2}{127} \)
(c) \( \frac{1}{126} \)
(d) \( \frac{1}{125} \)
Answer: (a) \( \frac{1}{128} \)

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

Question. If \( f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2 \sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix} \) then \( \lim_{x \to 0} \frac{f'(x)}{x} = \)
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (d) -2

Question. \( \lim_{x \to 0} \frac{\log \left[ \sec \left(\frac{x}{2}\right) \right]^{\cos x}}{\log [\sec x]^{\cos(x/2)}} = \)
(a) 14
(b) 15
(c) 16
(d) 17
Answer: (c) 16

Question. \( \lim_{x \to 0} \left[ \frac{100 \tan x \cdot \sin x}{x^2} \right] \) where [.] represents greatest integer function is
(a) 99
(b) 100
(c) 0
(d) 98
Answer: (a) 100

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. The graph of the function \( y = f(x) \) has a unique tangent at the point \( (e^a, 0) \) through which the graph passes then \( \lim_{x \to e^a} \frac{\log_e \{1 + 7f(x)\} - \sin f(x)}{3f(x)} \) is
(a) 1
(b) 2
(c) 0
(d) -1
Answer: (b) 2

Question. If [.] denotes the greatest integer function, then \( \lim_{x \to \frac{\pi}{2}} \left[ \frac{x - \frac{\pi}{2}}{\cos x} \right] = \)
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (d) -2

Question. If [.] denotes the greatest integer function then \( \lim_{x \to 0} \left[ \frac{x^2}{\tan x \cdot \sin x} \right] = \)
(a) 0
(b) 1
(c) -1
(d) does not exist
Answer: (a) 0

Question. \( \lim_{n \to \infty} \frac{1}{n^4} \sum_{r=1}^n r(r+2)(r+4) = \)
(a) \( \frac{3}{4} \)
(b) 0
(c) \( \frac{1}{8} \)
(d) \( \frac{1}{4} \)
Answer: (d) \( \frac{1}{4} \)

Question. \( \lim_{n \to \infty} \left[ \frac{7}{10} + \frac{29}{10^2} + \frac{133}{10^3} + \dots + \frac{5^n + 2^n}{10^n} \right] = \)
(a) 3/4
(b) 2
(c) 5/4
(d) 1/2
Answer: (c) 5/4

Question. Suppose \( f(n+1) = \frac{1}{2} \left\{ f(n) + \frac{9}{f(n)} \right\}, n \in N \). If \( f(n) > 0, \forall n \in N \), then \( \lim_{n \to \infty} f(n) = \)
(a) \( 3^{-1} \)
(b) \( -3^{-1} \)
(c) 3
(d) -3
Answer: (c) 3

Question. \( \lim_{x \to 0} [1^{1/\sin^2 x} + 2^{1/\sin^2 x} + \dots + n^{1/\sin^2 x}]^{\sin^2 x} = \)
(a) \( \infty \)
(b) 0
(c) \( \frac{n+1}{2} \)
(d) \( n \)
Answer: (d) \( n \)

Question. The value of \( \lim_{x \to 0} \left( \frac{e^{[\log(2^x-1)]^x} - (2^x - 1)^x \sin x}{e^{x \log x}} \right)^{1/x} \) is equal to
(a) \( e \)
(b) \( \frac{1}{e} \log 2 \)
(c) \( e \log 2 \)
(d) \( \log 2 \)
Answer: (b) \( \frac{1}{e} \log 2 \)

Question. \( \lim_{n \to \infty} \left(n^4 + n^3 + A_1 n^2 + A_2 n + A_3\right)^{1/2} - \left(n^4 + n^3 + B_1 n^2 + B_2 n + B_3\right)^{1/2} \) equals
(a) \( \frac{A_1 - B_1}{2} \)
(b) \( \frac{A_1 + B_3}{2} \)
(c) \( \frac{B_3 - A_1}{2} \)
(d) \( \frac{A_1 - B_3}{2} \)
Answer: (a) \( \frac{A_1 - B_1}{2} \)

Question. \( \lim_{n \to \infty} \frac{1 \cdot n^2 + 2(n-1)^2 + 3(n-2)^2 + \dots + n \cdot 1^2}{1^3 + 2^3 + \dots + n^3} \) equals
(a) 8/3
(b) 4/3
(c) 2/3
(d) 1/3
Answer: (d) 1/3

Question. \( \lim_{n \to \infty} \frac{1 - 2 + 3 - 4 + 5 - 6 + \dots - 2n}{\sqrt{n^2 + 1} + \sqrt{4n^2 - 1}} = \)
(a) 1/3
(b) -1/3
(c) -1/5
(d) 1/5
Answer: (b) -1/3

Question. If \( \lim_{x \to 0} (x^{-3} \sin 3x + a x^{-2} + b) \) exists and is equal to zero, then the value of \( a + 2b = \)
(a) 3
(b) 4
(c) 0
(d) 6
Answer: (d) 6

Question. The graph of \( y = f(x) \) has unique tangent at the point (a,0) through which the graph passes. Then \( \lim_{x \to a} \frac{\log [1 + 6 f(x)]}{3 f(x)} = \)
(a) 0
(b) 1
(c) 2
(d) \( \infty \)
Answer: (c) 2

Question. The value of \( \lim_{x \to 0} \left( 1 - \frac{1}{2^x} \right) \left( \frac{1}{\sqrt{\tan x + 4} - 2} \right) \) is
(a) log 16
(b) does not exist
(c) 3 log 2
(d) 6 log 2
Answer: (a) log 16

Question. The value of \( \lim_{x \to 0} \left\{ \left[ \frac{100x}{\sin x} \right] + \left[ \frac{99 \sin x}{x} \right] \right\} \), where [.] represents the greatest integer function, is
(a) 199
(b) 198
(c) 0
(d) 1
Answer: (b) 198

Question. \( \lim_{x \to 0^+} \frac{[x] + [x^2] + [x^3] + \dots + [x^{2n+1}] + n+1}{1 + [x^2] + [x] + 2x}, n \in N \) is equal to
(a) n + 1
(b) n
(c) 1
(d) 0
Answer: (d) 0

Question. If [.] denotes the greatest integer function, then \( \lim_{x \to 0} \frac{\tan([-2\pi^2]x^2) - x^2 \tan([-2\pi^2])}{\sin^2 x} = \)
(a) \( -20 + \tan 20 \)
(b) \( 20 + \tan 20 \)
(c) 20
(d) tan 20
Answer: (a) \( -20 + \tan 20 \)

Question. \( \lim_{n \to \infty} \frac{2^3-1}{2^3+1} \cdot \frac{3^3-1}{3^3+1} \cdot \dots \cdot \frac{n^3-1}{n^3+1} \) equals
(a) 1/3
(b) 2/3
(c) 1
(d) 3/2
Answer: (b) 2/3

Question. \( \lim_{n \to \infty} {}^n C_r \left(\frac{m}{n}\right)^r \left(1 - \frac{m}{n}\right)^{n-r} \) equals
(a) \( e^{-m} m^r \)
(b) \( \frac{m^r}{r!} \)
(c) \( \frac{m^r e^{-m}}{r!} \)
(d) \( \frac{e^{-r} r^m}{m!} \)
Answer: (c) \( \frac{m^r e^{-m}}{r!} \)

Question. The value of \( \lim_{x \to 0} \left\{ \sin^2 \left( \frac{\pi}{2 - ax} \right) \right\}^{\sec^2 \left( \frac{\pi}{2 - bx} \right)} \) is
(a) \( e^{-a/b} \)
(b) \( e^{-a^2/b^2} \)
(c) \( e^{2a/b} \)
(d) \( e^{4a/b} \)
Answer: (b) \( e^{-a^2/b^2} \)

Question. \( \lim_{x \to \infty} \left[ \frac{1^2}{1 - x^3} + \frac{3}{1 + x^2} + \frac{5^2}{1 - x^3} + \frac{7}{1 + x^2} + \dots \right] = \)
(a) -5/6
(b) -10/3
(c) 5/6
(d) 10/3
Answer: (b) -10/3

Question. Evaluate \( \lim_{n \to \infty} \left( 1 + \frac{1}{a_1} \right) \left( 1 + \frac{1}{a_2} \right) \dots \left( 1 + \frac{1}{a_n} \right) \) where \( a_1 = 1 \) and \( a_n = n(1 + a_{n-1}), \forall n \geq 2 \)
(a) 1/e
(b) \( 1/e^2 \)
(c) \( e \)
(d) 1
Answer: (c) \( e \)

Question. \( \lim_{n \to \infty} \frac{1}{n} \left\{ (n+1) \left(n+\frac{1}{2}\right) \left(n+\frac{1}{2^2}\right) \dots \left(n+\frac{1}{2^{n-1}}\right) \right\}^{1/n} = \)
(a) \( \frac{1}{2e^2} \)
(b) 1/e
(c) \( e^2 \)
(d) \( \frac{3}{2} e^2 \)
Answer: (c) \( e^2 \)

Question. Evaluate \( \lim_{n \to \infty} \left\{ \frac{1}{2} \tan \frac{x}{2} + \frac{1}{2^2} \tan \frac{x}{2^2} + \dots + \frac{1}{2^n} \tan \frac{x}{2^n} \right\} \)
(a) \( x \tan \frac{x}{2} \)
(b) \( \frac{1}{x} \cot \frac{x}{2} \)
(c) \( \frac{x - \cot x}{2} \)
(d) \( \frac{1}{x} - \cot x \)
Answer: (d) \( \frac{1}{x} - \cot x \)

Question. \( \lim_{x \to -\pi} \frac{|x + \pi|}{\sin x} = \)
(a) 1
(b) -1
(c) \( \pi \)
(d) does not exist
Answer: (d) does not exist

Question. If \( \ell(x) \) is least integer not less than x and \( g(x) \) is the greatest integer not greater than x then \( \lim_{x \to e + \pi} (\ell(x) + g(x)) = \)
(a) 1
(b) 9
(c) 11
(d) 13
Answer: (c) 11

Question. If \( 0 < P < 1 \) then \( \lim_{n \to \infty} \frac{n^P \sin^2(n!)}{n+1} = \)
(a) 0
(b) 1
(c) \( \infty \)
(d) 4/3
Answer: (a) 0

Question. \( \lim_{x \to a^+} \frac{\{x\} \sin(x-a)}{(x-a)^2} = \) where {x} denotes fractional part of x and \( a \in N \).
(a) 0
(b) 1
(c) a
(d) 5
Answer: (b) 1

Question. \( \lim_{x \to 0} \left\{ \left[ \frac{a \sin x}{x} \right] + \left[ \frac{b \tan x}{x} \right] \right\} = \) \( a, b \in N \), [where [ ] denotes G.I.F.]
(a) a + b
(b) a + b - 1
(c) 0
(d) \( \frac{a+b}{2} \)
Answer: (b) a + b - 1

Question. \( \lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4} = \)
(a) 1/6
(b) 1/5
(c) 1/4
(d) 1/2
Answer: (a) 1/6

Question. \( \lim_{x \to 0} \frac{1 - \sin[\cos x]}{[x] - [\sin x]} = \) (where [x] denotes greatest integral part of x)
(a) 0
(b) 1
(c) 2
(d) \( \infty \)
Answer: (d) \( \infty \)

Question. \( \lim_{x \to -1} \frac{1}{\sqrt{|x|} - \{-x\}} = \) (where {x} denotes fractional part of x)
(a) does not exist
(b) 1
(c) \( \infty \)
(d) 1/2
Answer: (a) does not exist

Question. \( \lim_{x \to 0} \left[ \frac{\sin(\text{sgn}(x))}{(\text{sgn}(x))} \right] = \) (where [x] denotes integral part of x)
(a) 0
(b) 1
(c) -1
(d) does not exist
Answer: (a) 0

Question. If \( f(x) = |x-1| - [x] \) (where [x] is greatest integer less than or equal to x) then.
(a) \( f(1^+) = -1; f(1^-) = 0 \)
(b) \( f(1^+) = 0 = f(1^-) \)
(c) \( \lim_{x \to 1} f(x) \) exits
(d) Cannot say any thing.
Answer: (a) \( f(1^+) = -1; f(1^-) = 0 \)