Class 11 Mathematics Limits And Derivatives MCQs Set 13

Question. \( \lim_{x \to 0} \frac{\sqrt[K]{1+x}-1}{x} \) (\( K \) is a positive integer)
(a) \( K \)
(b) \( -K \)
(c) \( \frac{1}{K} \)
(d) \( -\frac{1}{K} \)
Answer: (c) \( \frac{1}{K} \)

Question. \( \lim_{x \to 1} \frac{(2x-3)(\sqrt{x}-1)}{2x^2+x-3} = \)
(a) \( \frac{1}{10} \)
(b) \( -\frac{1}{10} \)
(c) \( \frac{2}{5} \)
(d) \( -\frac{2}{5} \)
Answer: (b) \( -\frac{1}{10} \)

Question. \( \lim_{x \to 0} \frac{\sqrt[3]{1+\sin x}-\sqrt[3]{1-\sin x}}{x} = \)
(a) 0
(b) 1
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (c) \( \frac{2}{3} \)

Question. If \( \lim_{x \to 5} \frac{x^k-5^k}{x-5} = 500 \), then the positive integral value of \( k \) is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. \( \lim_{x \to 1} \frac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}} = \)
(a) \( 1+\frac{1}{\sqrt{2}} \)
(b) \( 1-\frac{1}{\sqrt{2}} \)
(c) \( -1+\frac{1}{\sqrt{2}} \)
(d) \( -1-\frac{1}{\sqrt{2}} \)
Answer: (a) \( 1+\frac{1}{\sqrt{2}} \)

Question. If \( a > 0 \) and \( \lim_{x \to a} \frac{a^x-x^a}{x^x-a^a} = -1 \) then \( a = \)
(a) 0
(b) 1
(c) \( e \)
(d) \( 2e \)
Answer: (b) 1

Question. \( \lim_{x \to 0} \frac{x(1-\sqrt{1-x^2})}{\sqrt{1-x^2}(\sin^{-1}(x))^3} = \)
(a) 1
(b) \( \frac{1}{2} \)
(c) \( -\frac{1}{2} \)
(d) -1
Answer: (b) \( \frac{1}{2} \)

Question. If \( \lim_{x \to 0} \frac{\cos 4x + a \cos 2x + b}{x^4} \) is finite then the value of \( a, b \) respectively
(a) 5
(b) -5, -4
(c) -4, 3
(d) 4, 5
Answer: (c) -4, 3

Question. \( \lim_{x \to 1} \left[ \sec\left(\frac{\pi x}{2}\right) \log x \right] \) is
(a) \( -\frac{2}{\pi} \)
(b) \( -\frac{\pi}{2} \)
(c) \( \frac{2}{\pi} \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( -\frac{2}{\pi} \)

Question. \( \lim_{x \to 0} \frac{\sqrt{1+x^2}-\sqrt{1-x+x^2}}{3^x-1} = \)
(a) \( \log 9 \)
(b) \( \frac{1}{\log 9} \)
(c) \( \log 3 \)
(d) \( \frac{1}{\log 3} \)
Answer: (b) \( \frac{1}{\log 9} \)

Question. \( \lim_{x \to 0} \frac{\sin x}{\sqrt{x^2}} = \)
(a) 1
(b) -1
(c) 0
(d) doesn’t exist
Answer: (d) doesn’t exist

Question. \( \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[\frac{x}{3}\right]^3 \right) \) is (where \( [ ] \) is g.i.f)
(a) 0
(b) \( \frac{64}{27} \)
(c) \( \frac{8}{3} \)
(d) \( \frac{10}{3} \)
Answer: (c) \( \frac{8}{3} \)

Question. If \( f : R \to R \) is defined by (where \( [  ] \) is g.i.f) \( f(x) = [x-3] + |x-4| \) for \( x \in R \) then \( \lim_{x \to 3^-} f(x) = \) 
(a) -2
(b) -1
(c) 0
(d) 2
Answer: (c) 0

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

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

Question. \( \lim_{x \to 1} (1-x) \tan \left( \frac{\pi x}{2} \right) = \)
(a) \( \pi \)
(b) \( 2\pi \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{2}{\pi} \)
Answer: (d) \( \frac{2}{\pi} \)

Question. \( \lim_{x \to \frac{\pi}{6}} \frac{3 \sin x - \sqrt{3} \cos x}{6x - \pi} = \)
(a) \( \sqrt{3} \)
(b) \( \frac{1}{\sqrt{3}} \)
(c) \( -\sqrt{3} \)
(d) \( -\frac{1}{\sqrt{3}} \)
Answer: (b) \( \frac{1}{\sqrt{3}} \)

Question. \( \lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\frac{\pi}{2} - x)^3} = \)
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{2} \)
(c) 2
(d) -2
Answer: (b) \( \frac{1}{2} \)

Question. \( \lim_{x \to 0} \frac{\sec 4x - \sec 2x}{\sec 3x - \sec x} = \)
(a) \( \frac{3}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{1}{3} \)
(d) \( \frac{3}{4} \)
Answer: (a) \( \frac{3}{2} \)

Question. \( \lim_{x \to 0} \frac{3 \sin (x^g) - \sin(3x^g)}{x^3} = \)
(a) \( \left( \frac{\pi}{200} \right)^3 \)
(b) \( 4 \left( \frac{\pi}{200} \right)^3 \)
(c) \( \frac{\pi}{200} \)
(d) \( \frac{\pi}{100} \)
Answer: (b) \( 4 \left( \frac{\pi}{200} \right)^3 \)

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

Evaluation of exponential & logarithmic limits :

Question. \( \lim_{x \to 0} \log_{10} \frac{\log(1+x)}{x} = \)
(a) 0
(b) 1
(c) \( e \)
(d) \( 1/e \)
Answer: (a) 0

Question. The value of \( \lim_{x \to 2} \frac{2^x + 2^{3-x} - 6}{\sqrt{2^{-x}} - 2^{1-x}} \) is
(a) 16
(b) 8
(c) 4
(d) 2
Answer: (b) 8

Question. \( \lim_{x \to 0} \frac{e^x + \sin x - 1}{\log(1+x)} = \)
(a) 1
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{3} \)
(d) 2
Answer: (d) 2

Question. \( \lim_{x \to 0} \frac{10^x - 2^x - 5^x + 1}{x \tan x} \) is
(a) \( \log 2 \)
(b) \( \frac{\log 2}{\log 5} \)
(c) \( (\log 2)(\log 5) \)
(d) \( \log 10 \)
Answer: (c) \( (\log 2)(\log 5) \)

Question. \( \lim_{n \to \infty} \left( \frac{1}{a(a+d)} + \frac{1}{(a+d)(a+2d)} + \frac{1}{(a+2d)(a+3d)} + .... \text{ to } n \text{ terms} \right) \) (\( a > 0, d > 0 \)) =
(a) \( \frac{1}{a} \)
(b) \( \frac{1}{d} \)
(c) \( \frac{a}{d} \)
(d) \( \frac{1}{ad} \)
Answer: (d) \( \frac{1}{ad} \)

Question. \( \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \frac{1}{4} + .... + \frac{1}{2^n}}{1 + \frac{1}{3} + \frac{1}{9} + .... + \frac{1}{3^n}} = \)
(a) 4/3
(b) 3/4
(c) 1/2
(d) 0
Answer: (a) 4/3

Question. \( \lim_{x \to \infty} \frac{x^2 \sin \left(\frac{1}{x}\right) - x}{1 - |x|} = \)
(a) 0
(b) 1
(c) –1
(d) 2
Answer: (a) 0

Question. If \( 0 < h < q \) then \( \lim_{n \to \infty} (q^n + h^n)^{1/n} = \)
(a) \( e \)
(b) \( h \)
(c) \( q \)
(d) 0
Answer: (c) \( q \)

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

Question. \( \lim_{x \to \infty} x \cos \left( \frac{\pi}{8x} \right) \sin \left( \frac{\pi}{8x} \right) = \)
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{8} \)
(d) \( \frac{\pi}{4} \)
Answer: (c) \( \frac{\pi}{8} \)

Question. \( \lim_{x \to \infty} (\sqrt{x^2 + ax + a^2} - \sqrt{x^2 + a^2}) = \)
(a) 0
(b) \( \frac{a}{2} \)
(c) \( -\frac{a}{2} \)
(d) \( a \)
Answer: (b) \( \frac{a}{2} \)

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

Question. For \( a > 1 \) then \( \lim_{x \to \infty} \frac{a^x - a^{-x}}{a^x + a^{-x}} = \)
(a) 1
(b) 1/2
(c) 1/3
(d) 1/15
Answer: (a) 1

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

Question. \( \lim_{x \to \infty} \frac{8|x| + 3x}{3|x| - 2x} = \)
(a) 11
(b) 8
(c) 0
(d) 1/8
Answer: (a) 11

Question. \( \lim_{x \to \infty} \frac{(x+1)^{10} + (x+2)^{10} + ..... + (x+100)^{10}}{x^{10} + 10^{10}} = \)
(a) 10
(b) 100
(c) 1000
(d) 1
Answer: (b) 100

Question. \( \lim_{n \to \infty} \frac{3 \cdot 2^{n+1} - 4 \cdot 5^{n+1}}{5 \cdot 2^n + 7 \cdot 5^n} = \)
(a) -20/7
(b) 20/7
(c) 10/7
(d) –10/7
Answer: (a) -20/7

Question. \( \lim_{n \to \infty} \frac{1}{n^3} \left\{ 1 + 3 + 6 + ........... + \frac{n(n+1)}{2} \right\} = \)
(a) 0
(b) 2
(c) 1/6
(d) 1/3
Answer: (c) 1/6

Question. \( \lim_{x \to 1} (2-x)^{\tan \left( \frac{\pi x}{2} \right)} = \)
(a) \( e^{1/\pi} \)
(b) \( e^{2/\pi} \)
(c) \( -e^{2/\pi} \)
(d) \( e \)
Answer: (b) \( e^{2/\pi} \)