Class 11 Mathematics Limits And Derivatives MCQs Set 17

Question. \( \text{Lt}_{x \to 0} \frac{(1+x)^3 - 1 - 3x}{(1+x)^2 - 1 - 2x} = \)
(a) 2
(b) 3
(c) 5
(d) -2
Answer: (b) 3

Question. If \( f(x) = \frac{4-7x}{7x+4} \), \( \text{Lt}_{x \to 0} f(x) = l \) and \( \text{Lt}_{x \to \infty} f(x) = m \) the quadratic equation having roots as \( \frac{1}{l} \) and \( \frac{1}{m} \) is
(a) \( x^2 - 1 = 0 \)
(b) \( x^2 + 1 = 0 \)
(c) 1/2
(d) \( x^3 - 1 = 0 \)
Answer: (a) \( x^2 - 1 = 0 \)

Question. \( \text{Lim}_{x \to 2} \frac{\sqrt{x+7}-3\sqrt{2x-3}}{\sqrt[3]{x+6}-2\sqrt[3]{3x-5}} = \)
(a) 34/23
(b) 23/17
(c) 7/23
(d) 23/7
Answer: (a) 34/23

Question. \( \text{Lim}_{x \to 1} \left[ \frac{x^3 + 2x^2 + x + 1}{x^2 + 2x + 3} \right]^{\frac{1-\cos(x-1)}{(x-1)^2}} = \)
(a) e
(b) \( e^{1/2} \)
(c) 1
(d) \( (5/6)^{1/2} \)
Answer: (d) \( (5/6)^{1/2} \)

Question. \( \text{Lim}_{x \to 0} \frac{x\sqrt{y^2-(y-x)^2}}{\left\{ \sqrt{(8xy-4x^2)} + \sqrt{8xy} \right\}^3} = \)
(a) \( \frac{1}{4y} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{2\sqrt{2}} \)
(d) \( \frac{1}{128y} \)
Answer: (d) \( \frac{1}{128y} \)

Question. \( \text{Lt}_{x \to \infty} \frac{x^5}{5^x} = \)
(a) 0
(b) 1
(c) \( \infty \)
(d) does not exist
Answer: (a) 0

Question. \( \text{lim}_{x \to 8} \frac{\sqrt{1+\sqrt{1+x}}-2}{x-8} = \) 
(a) 3/2
(b) 1/4
(c) 1/24
(d) 1/5
Answer: (c) 1/24

Question. \( \text{lim}_{x \to 2} \frac{\sqrt{1-\cos\{2(x-2)\}}}{x-2} \)
(a) equals \( \sqrt{2} \)
(b) equals \( -\sqrt{2} \)
(c) equals \( \frac{1}{\sqrt{2}} \)
(d) does not exist
Answer: (d) does not exist

Question. If \( l_1 = \text{lim}_{x \to 2^+} (x+[x]) \), \( l_2 = \text{lim}_{x \to 2^-} (2x-[x]) \) and \( l_3 = \text{lim}_{x \to \frac{\pi}{2}} \frac{\cos x}{x-\frac{\pi}{2}} \) then [where [ ] denotes G.I.F.]
(a) \( l_1 < l_2 < l_3 \)
(b) \( l_2 < l_3 < l_1 \)
(c) \( l_3 < l_2 < l_1 \)
(d) \( l_1 < l_3 < l_2 \)
Answer: (c) \( l_3 < l_2 < l_1 \)

Question. If \( a = \min \{x^2 + 4x + 5, x \in R\} \) and \( b = \text{Lim}_{\theta \to 0} \frac{1-\cos 2\theta}{\theta^2} \) then the value of \( \sum_{r=0}^{n} a^r b^{n-r} = \)
(a) \( \frac{2^{n+1}-1}{4.2^n} \)
(b) \( 2^{n+1}-1 \)
(c) \( \frac{2^{n+1}-1}{3.2^n} \)
(d) \( 2^n - 1 \)
Answer: (b) \( 2^{n+1}-1 \)

Question. \( \text{lim}_{x \to 0} \frac{\tan x - \sin x}{x^2} = \) 
(a) 0
(b) 1
(c) 1/2
(d) -1/2
Answer: (a) 0

Question. \( \text{lim}_{x \to 0} \frac{(1-\cos 2x)(3 + \cos x)}{x \tan 4x} = \) 
(a) -1/4
(b) 1/2
(c) 1
(d) 2
Answer: (d) 2

Question. \( \text{lim}_{x \to 0} \frac{\tan^3 x - \sin^3 x}{x^5} = \) 
(a) 5/2
(b) 3/2
(c) 3/5
(d) 2/5
Answer: (b) 3/2

Question. \( \text{Lt}_{x \to 0} \frac{\text{Sin} 2x + 2\text{Sin}^2 x - 2\text{Sinx}}{\text{Cosx} - \text{Cos}^2 x} = \)
(a) 0
(b) 1
(c) 2/3
(d) 4
Answer: (d) 4

Question. \( \text{Lt}_{x \to 0} \frac{3\sin x - \sin 3x}{x \cdot \tan^2 2x} = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (b) 1

Question. \( \text{Lim}_{x \to 0} \frac{8}{x^8} \left[ 1 - \cos \frac{x^2}{2} - \cos \frac{x^2}{4} + \cos \frac{x^2}{2} \cdot \cos \frac{x^2}{4} \right] = \)
(a) 1/16
(b) 1/15
(c) 1/32
(d) 1
Answer: (c) 1/32

Question. Arrange the following limits in the ascending order.
1) \( \text{Lim}_{x \to 0} \frac{\tan^4 x - \sin^4 x}{x^6} \)
2) \( \text{Lim}_{x \to 0} \frac{\tan^8 x - \sin^8 x}{x^5 \tan x^5} \)
3) \( \text{Lim}_{x \to 0} \frac{\tan^3 x - \sin^3 x}{x \sin^4 x} \)
4) \( \text{Lim}_{x \to 0} \frac{\tan^5 x - \sin^5 x}{x^2 \cdot \sinh^3 x \cdot \tan^2 x} \)
(a) 1, 2, 3, 4
(b) 3, 1, 4, 2
(c) 1, 2, 4, 3
(d) 2, 1, 3, 4
Answer: (b) 3, 1, 4, 2

Question. The value of \( \theta \), is \( \text{lim}_{\theta \to 0} \frac{\cos^2 \{1-\cos^2(1-\cos^2 \dots (1-\cos^2 \theta)) \dots \}}{\sin \left( \frac{\pi(\sqrt{\theta+4}-2)}{\theta} \right)} \)
(a) \( \frac{\sqrt{2}}{4} \)
(b) \( \sqrt{2} \)
(c) 1
(d) 2
Answer: (b) \( \sqrt{2} \)

Question. \( \text{Lt}_{x \to \pi} \frac{\sqrt{2+\cos x}-1}{(\pi-x)^2} = \)
(a) 0
(b) 1/4
(c) 1/2
(d) 2
Answer: (b) 1/4

Question. If \( \text{lim}_{x \to 0} \frac{\{(a-n)nx - \tan x\}\sin nx}{x^2} = 0 \), where \( n \) is a non-zero real number, then 'a' =
(a) 0
(b) \( \frac{n+1}{n} \)
(c) n
(d) \( n + \frac{1}{n} \)
Answer: (d) \( n + \frac{1}{n} \)

Question. \( \text{Lim}_{h \to 0} \left[ \frac{\sqrt{3} \sin(\frac{\pi}{6}+h) - \cos(\frac{\pi}{6}+h)}{\sqrt{3}h(\sqrt{3}\cosh - \sinh)} \right] = \)
(a) \( -\frac{2}{\sqrt{3}} \)
(b) \( -\frac{4}{3} \)
(c) \( \frac{2}{\sqrt{3}} \)
(d) \( \frac{4}{3} \)
Answer: (d) \( \frac{4}{3} \)

Question. The value of \( \text{lim}_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)} \) is
(a) 1
(b) -1
(c) 0
(d) 2
Answer: (a) 1

Question. Arrange the following limits in the ascending order
1) \( \text{lim}_{x \to \infty} \left( \frac{1+x}{2+x} \right)^{x+2} \)
2) \( \text{lim}_{x \to 0} (1+2x)^{3/x} \)
3) \( \text{lim}_{\theta \to 0} \frac{\sin \theta}{2\theta} \)
4) \( \text{lim}_{x \to 0} \frac{\log_e(1+x)}{x} \)
(a) 1, 2, 3, 4
(b) 1, 3, 4, 2
(c) 1, 4, 3, 2
(d) 3, 4, 1, 2
Answer: (b) 1, 3, 4, 2

Question. \( \text{lim}_{x \to 0} \frac{(4^x - 1)^3}{\sin(\frac{x}{4}) \log_e(1 + \frac{x^2}{3})} = \)
(a) \( (\log_e 4)^3 \)
(b) \( \log_e 4 \)
(c) \( 12(\log_e 4)^3 \)
(d) \( 5(\log_e 4)^3 \)
Answer: (c) \( 12(\log_e 4)^3 \)

Question. \( \text{lim}_{x \to 0} \frac{e^{1/x}-1}{e^{1/x}+1} = \)
(a) 1
(b) -1
(c) 0
(d) does not exist
Answer: (d) does not exist

Question. \( \text{lim}_{n \to \infty} \frac{1^3+2^3+3^3+\dots+n^3}{3n^4+5n^3+6} = \)
(a) 1/3
(b) 1/5
(c) 1/6
(d) 1/12
Answer: (d) 1/12

Question. \( \text{lim}_{x \to \infty} \frac{3\sqrt{x} + 5\sin^2 x - 10\log x}{5\sqrt{x} + 7\cos^2 x + 100\log x} = \)
(a) 3/5
(b) 5/3
(c) 15
(d) 1/15
Answer: (a) 3/5

Question. \( \text{lim}_{n \to \infty} \frac{(\sqrt{n^2+1}+n)^2}{\sqrt[3]{n^6+1}} = \)
(a) 1
(b) 1/2
(c) 1/3
(d) 4
Answer: (d) 4

Question. \( \text{lim}_{x \to \infty} \frac{(2+x)^{20}(4+x)^3}{(2-x)^{23}} = \)
(a) -1
(b) 1
(c) 6
(d) 2
Answer: (a) -1

Question. \( \text{lim}_{n \to \infty} \cos(\pi \sqrt{n^2+n}) \) in equal to
(a) 0
(b) 1
(c) 2
(d) does not exist
Answer: (a) 0

Question. \( \text{lim}_{n \to \infty} \left( \frac{1}{5} \right)^{\log_{1/5} (\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+ \dots \text{ to } n \text{ terms})} \) equals
(a) 2
(b) 4
(c) 8
(d) 0
Answer: (b) 4

Question. \( \text{lim}_{n \to \infty} \frac{1^4+2^4+3^4+\dots+n^4}{n^5} - \text{lim}_{n \to \infty} \frac{1^3+2^3+3^3+\dots+n^3}{n^5} \) is
(a) 1/5
(b) 2/5
(c) 3/5
(d) 4/5
Answer: (a) 1/5

Question. If \( a > 1 \), then the value of \( \text{lim}_{x \to \infty} \frac{a^{\sqrt{x}}-a^{1/\sqrt{x}}}{a^{\sqrt{x}}+a^{1/\sqrt{x}}} \) is
(a) 1
(b) 0
(c) 2
(d) 3
Answer: (a) 1

Question. for \( x > 0 \); \( \text{lim}_{x \to 0} \left( (\sin x)^{1/x} + (\frac{1}{x})^{\sin x} \right) = \)
(a) 0
(b) -1
(c) 1
(d) 2
Answer: (c) 1

Question. \( \text{lim}_{x \to 0} \left( \frac{1+\tan x}{1+\sin x} \right)^{\text{cosec} x} \) is equal to
(a) 1/e
(b) e
(c) \( e^2 \)
(d) 1
Answer: (d) 1

Question. \( \text{lim}_{x \to \infty} \left( \frac{x+6}{x+1} \right)^{x+4} = \) 
(a) \( e^4 \)
(b) \( e^6 \)
(c) \( e^5 \)
(d) e
Answer: (c) \( e^5 \)

Question. \( \text{lim}_{x \to \infty} \left( \frac{x+5}{x+2} \right)^{x+3} = \) 
(a) e
(b) \( e^2 \)
(c) \( e^3 \)
(d) \( e^5 \)
Answer: (c) \( e^3 \)

Question. If \( f'(0) = 3 \), then \( \text{lim}_{x \to 0} \frac{x^2}{f(x^2)-6f(4x^2)+5f(7x^2)} = \)
(a) 1/36
(b) -1/36
(c) 1/34
(d) 1/106
Answer: (a) 1/36

Question. \( \text{lim}_{x \to 0} \frac{(1+x)^{1/x} - e}{x} = \)
(a) 1
(b) e/2
(c) -e/2
(d) 2/e
Answer: (c) -e/2

Question. If [x] denotes the greatest integer less than or equal to x then \( \text{lim}_{n \to \infty} \frac{1}{n^2} \left[ [1^2x] + [2^2x] + [3^2x] + \dots + [n^2x] \right] = \)
(a) x/2
(b) x/3
(c) x/6
(d) 0
Answer: (b) x/3