Class 11 Mathematics Limits And Derivatives MCQs Set 14

Question. \( \operatorname{Lim}_{x \rightarrow a} \frac{x^{5/8} - a^{5/8}}{x^{1/3} - a^{1/3}} = \)
(a) \( \frac{15}{8} a^{7/24} \)
(b) \( \frac{15}{4} a^{7/24} \)
(c) \( -\frac{15}{8} a^{7/24} \)
(d) \( -\frac{15}{4} a^{7/24} \)
Answer: (a) \( \frac{15}{8} a^{7/24} \)

Question. \( \operatorname{Lim}_{x \rightarrow 2} \frac{7x^2 - 11x - 6}{3x^2 - x - 10} = \)
(a) \( \frac{17}{11} \)
(b) \( \frac{11}{17} \)
(c) \( \frac{17}{14} \)
(d) \( -\frac{17}{11} \)
Answer: (a) \( \frac{17}{11} \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{(1+x)^m - (1-x)^m}{(1+x)^n - (1-x)^n} = \)
(a) 1
(b) \( \frac{m+n}{m-n} \)
(c) \( \frac{n}{m} \)
(d) \( \frac{m}{n} \)
Answer: (c) \( \frac{n}{m} \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{\sqrt{4+x} - \sqrt[3]{8+3x}}{x} = \)
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{2} \)
(c) -3
(d) 0
Answer: (d) 0

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{\sqrt{1+\operatorname{Tan}x} - \sqrt{1-\operatorname{Tan}x}}{\sin x} = \)
(a) 0
(b) 1
(c) 2
(d) \( \frac{1}{2} \)
Answer: (b) 1

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

Question. If \( 7 - \frac{x^2}{12} \le f(x) \le 7 + \frac{x^3}{5} \) for all \( x \ne 0 \), then \( \operatorname{Lim}_{x \rightarrow 0} f(x) = \)
(a) 3
(b) 5
(c) 7
(d) 9
Answer: (c) 7

Question. If \( f(x) = \begin{cases} \frac{x^2}{2}, & \text{if } 0 \le x < 1 \\ 2x^2 - 2x + \frac{3}{2}, & \text{if } 1 \le x \le 2 \end{cases} \), then \( \lim_{x \rightarrow 1} f(x) = \)
(a) 1/2
(b) 3/2
(c) does not exist
(d) -1/2
Answer: (c) does not exist

Question. \( \lim_{x \rightarrow 0} \frac{\sin[\cos x]}{1+[\cos x]} \) is (where [ ] is g.i.f)
(a) 1
(b) 0
(c) does not exist
(d) 2
Answer: (b) 0

Question. \( \operatorname{Lim}_{x \rightarrow 0} \left( \frac{\sec ax - \sec bx}{x^2} \right) = \)
(a) \( \frac{a^2 - b^2}{2} \)
(b) \( \frac{b^2 - a^2}{2} \)
(c) 0
(d) 1
Answer: (b) \( \frac{b^2 - a^2}{2} \)

Question. \( \operatorname{Lim}_{x \rightarrow \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( \frac{\pi}{4} - x \right) (\cos x + \sin x)} = \)
(a) 2
(b) 1
(c) 0
(d) 3
Answer: (b) 1

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{\sin x \sin \left( \frac{\pi}{3} + x \right) \sin \left( \frac{\pi}{3} - x \right)}{x} = \)
(a) 3/4
(b) 1/4
(c) 4/3
(d) 0
Answer: (a) 3/4

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{(1 - \cos 2x) \sin 5x}{x^2 \sin 3x} = \)
(a) 10/3
(b) 3/10
(c) 6/5
(d) 56
Answer: (a) 10/3

Question. \( \operatorname{Lim}_{x \rightarrow 5} \frac{\sin^2(x-5) \tan(x-5)}{(x^2-25)(x-5)} = \)
(a) 1
(b) 1/10
(c) 0
(d) -6
Answer: (c) 0

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{1 - \cos 2x^\circ}{x^2} = \)
(a) 0
(b) 1
(c) \( \left( \frac{\pi}{180} \right)^2 \)
(d) \( 2 \left( \frac{\pi}{180} \right)^2 \)
Answer: (d) \( 2 \left( \frac{\pi}{180} \right)^2 \)

Question. \( \lim_{x \rightarrow 0} \frac{1 - \cos x}{x \log(1+x)} = \)
(a) 1
(b) 0
(c) -1
(d) 1/2
Answer: (d) 1/2

Question. \( \operatorname{Lt}_{x \rightarrow 0} \frac{1}{x^8} \left[ 1 - \cos\left(\frac{x^2}{2}\right) \right] \left[ 1 - \cos\left(\frac{x^2}{4}\right) \right] = \)
(a) \( \frac{1}{8} \)
(b) \( \frac{1}{8^2} \)
(c) \( \frac{2}{8^3} \)
(d) \( \frac{2}{8^4} \)
Answer: (c) \( \frac{2}{8^3} \)

Question. \( \operatorname{Lt}_{x \rightarrow \frac{\pi}{4}} \frac{1 - \cot^3 x}{2 - \cot x - \cot^3 x} = \)
(a) 11/4
(b) 3/4
(c) 1/2
(d) 0
Answer: (b) 3/4

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{\log_e(1+x)}{3^x - 1} = \)
(a) \( \log_e 3 \)
(b) 0
(c) 1
(d) \( \log_3 e \)
Answer: (d) \( \log_3 e \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{e^{\alpha x} - e^{\beta x}}{\sin \alpha x - \sin \beta x} = \)
(a) 0
(b) 1/2
(c) 1/3
(d) 1
Answer: (d) 1

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{e^{2x} - 1}{3x} = \)
(a) 2/3
(b) 6
(c) 3/2
(d) 1/6
Answer: (a) 2/3

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{\log(1+ax) - \log(1+bx)}{x} = \)
(a) \( a + b \)
(b) \( a - b \)
(c) \( -(a + b) \)
(d) \( ab \)
Answer: (b) \( a - b \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{4^x - 9^x}{x(4^x + 9^x)} = \)
(a) \( \log \left( \frac{3}{2} \right) \)
(b) \( \log \left( \frac{2}{3} \right) \)
(c) \( \log \left( \frac{4}{3} \right) \)
(d) \( \log 2 \)
Answer: (b) \( \log \left( \frac{2}{3} \right) \)

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{a^x - 1}{b^x - 1} = \)
(a) \( \log_b a \)
(b) \( \log_a b \)
(c) \( \log_e ab \)
(d) \( \log_e \left( \frac{a}{b} \right) \)
Answer: (a) \( \log_b a \)

Question. \( \operatorname{Lim}_{x \rightarrow \infty} 2x \left( \sqrt{x^2+1} - x \right) = \)
(a) 1
(b) 1/2
(c) 0
(d) -1
Answer: (a) 1

Question. \( \operatorname{Lim}_{x \rightarrow \infty} \frac{\sqrt[7]{x^7-1} + \sqrt[5]{x^5+2} + \sqrt[9]{x^9-2}}{\sqrt[6]{x^6+1} + \sqrt[5]{x^5+1} - \sqrt[4]{x^4+4}} = \)
(a) 3
(b) – 3
(c) 1/3
(d) – 1/3
Answer: (a) 3

Question. \( \operatorname{Lt}_{x \rightarrow \infty} 5^x \sin\left(\frac{a}{5^x}\right) = \)
(a) 1
(b) a
(c) \( \infty \)
(d) not defined
Answer: (b) a

Question. \( \operatorname{Lim}_{n \rightarrow \infty} \frac{2^{n+1} - 1}{2^n + 1} = \)
(a) 1
(b) 2
(c) -1
(d) 0
Answer: (b) 2

Question. \( \operatorname{Lim}_{n \rightarrow \infty} \frac{(1+2+3 + \dots + nterms)(1^2+2^2 + \dots + nterms)}{n(1^3+2^3 + \dots + nterms)} = \)
(a) 3/2
(b) 2/3
(c) 1
(d) 0
Answer: (b) 2/3

Question. \( \operatorname{Lim}_{n \rightarrow \infty} \left( \frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \dots + \frac{1}{(2n-1)(2n+1)} \right) = \)
(a) 1
(b) 1/2
(c) 1/3
(d) 1/4
Answer: (b) 1/2

Question. \( \operatorname{Lim}_{n \rightarrow \infty} \frac{1+3+5 + \dots + (2n-1)}{2+4+6 + \dots + 2n} = \)
(a) 0
(b) 1
(c) -1
(d) 5
Answer: (b) 1

Question. \( \operatorname{Lim}_{n \rightarrow \infty} \frac{(n+2)! + (n+1)!}{(n+3)!} = \)
(a) 1
(b) 1/2
(c) 0
(d) .2
Answer: (c) 0

Question. If \( \operatorname{Lt}_{x \rightarrow \infty} \left( 1 + \frac{\lambda}{x} + \frac{\mu}{x^2} \right)^{2x} = e^2 \) then
(a) \( \lambda = 1, \mu = 2 \)
(b) \( \lambda = 2, \mu = 1 \)
(c) \( \lambda = 1 \), \( \mu = \text{any real constant} \)
(d) \( \lambda = \mu = 1 \)
Answer: (c) \( \lambda = 1 \), \( \mu = \text{any real constant} \)

Question. \( \operatorname{Lim}_{x \rightarrow \infty} \left( \frac{x+a}{x+b} \right)^{x+b} = \)
(a) 1
(b) \( e^{b-a} \)
(c) \( e^{a-b} \)
(d) \( e^b \)
Answer: (c) \( e^{a-b} \)

Question. \( \operatorname{Lim}_{x \rightarrow \pi} (1 - 4 \tan x)^{\cot x} = \)
(a) e
(b) \( e^4 \)
(c) \( e^{-1} \)
(d) \( e^{-4} \)
Answer: (d) \( e^{-4} \)

Question. \( \operatorname{Lim}_{x \rightarrow \infty} \left( \frac{x^2+1}{x^2-1} \right)^{x^2} = \)
(a) e
(b) 1/e
(c) \( e^2 \)
(d) \( e^{-2} \)
Answer: (c) \( e^2 \)

Question. \( \operatorname{Lim}_{x \rightarrow \infty} \left( \frac{3x-4}{3x+2} \right)^{\frac{x+1}{3}} = \)
(a) \( e^{-2/3} \)
(b) \( e^{3/2} \)
(c) \( e^{2/3} \)
(d) e
Answer: (a) \( e^{-2/3} \)

Question. \( \operatorname{Lim}_{x \rightarrow 1} x^{\left( \frac{1}{1-x^2} \right)} = \)
(a) \( e^{-1/2} \)
(b) \( e^{2/3} \)
(c) \( e^{-3/2} \)
(d) e
Answer: (a) \( e^{-1/2} \)

Question. \( \operatorname{Lim}_{x \rightarrow 1} \left( \frac{1+x}{2+x} \right)^{\left( \frac{1-\sqrt{x}}{1-x} \right)} = \)
(a) 2/3
(b) \( \sqrt{\frac{2}{3}} \)
(c) \( \sqrt{\frac{3}{2}} \)
(d) \( \frac{\sqrt{2}}{3} \)
Answer: (b) \( \sqrt{\frac{2}{3}} \)

Question. If \( f(x) = x^2 + x + 1 \), then \( \operatorname{Lim}_{x \rightarrow 1} \frac{f(x) - f(1)}{x-1} = \)
(a) 3
(b) 0
(c) -1
(d) 2
Answer: (a) 3