Class 11 Mathematics Limits And Derivatives MCQs Set 16

Question. \( \operatorname{Lim}_{x \rightarrow 1} \frac{\left(\sum_{K=1}^{200} x^{K}\right)-200}{x-1}= \)
(a) 5050
(b) 1000
(c) 2010
(d) 20100
Answer: (d) 20100

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

Question. Let \( \alpha \) and \( \beta \) be the roots of \( a x^{2}+b x+c=0 \), then \( \operatorname{Lt}_{x \rightarrow \alpha} \frac{1-\cos \left(a x^{2}+b x+c\right)}{(x-\alpha)^{2}}= \)
(a) \( \frac{a^{2}(\alpha-\beta)^{2}}{2} \)
(b) \( \frac{a^{2}}{2(\alpha-\beta)^{2}} \)
(c) \( \frac{a^{2}}{(\alpha-\beta)^{2}} \)
(d) \( -\frac{a^{2}}{2(\alpha-\beta)^{2}} \)
Answer: (a) \( \frac{a^{2}(\alpha-\beta)^{2}}{2} \)

Question. If \( f(x)=\left\{\begin{array}{l}\sin x, x \neq n \pi, n \in Z \\ 2, \text { otherwise }\end{array}\right. \) and \( g(x)=\left\{\begin{array}{ll}x^{2}+1, & x \neq 0,2 \\ 4, & x=0 \\ 5, & x=2\end{array}\right. \) then \( \lim _{x \rightarrow 0} g[f(x)]= \)
(a) 2
(b) 4
(c) 5
(d) 1
Answer: (d) 1

Question. \( \lim _{y \rightarrow x} \frac{y^{y}-x^{x}}{y-x}= \)
(a) \( x y^{x-1} \)
(b) \( x^{x}(1+\log x) \)
(c) \( y \cdot x^{y-1} \)
(d) \( x y \)
Answer: (b) \( x^{x}(1+\log x) \)

Question. If \( \lim _{x \rightarrow \infty}\left(\frac{x^{2}-1}{x+1}-a x-b\right)=2 \), then
(a) \( a=1 \) and \( b=-3 \)
(b) \( a=1 \) and \( b=2 \)
(c) \( a=0 \) and \( b=-1 \)
(d) \( a=2 \) and \( b=1 \)
Answer: (a) \( a=1 \) and \( b=-3 \)

Question. Let \( \{x\} \) denote the fractional part of \( x \). Then \( \lim _{x \rightarrow 0} \frac{\{x\}}{\tan \{x\}} \) is equal to
(a) \( -1 \)
(b) \( 0 \)
(c) \( 1 \)
(d) Does not exist
Answer: (d) Does not exist

Question. \( \operatorname{Lt}_{x \rightarrow 4^{+}} \frac{x^{2}-7 x+12}{x-[x]}= \) [where [ ] denotes G.I.F.]
(a) 1
(b) 0
(c) 2
(d) does not exists
Answer: (a) 1

Question. If \( f: R \rightarrow R \) defined by \( f(x)=\left\{\begin{array}{ll}\frac{x-2}{x^{2}-3 x+2} & \text { if } x \in R-\{1,2\} \\ 2 & \text { if } x=1 \\ 1 & \text { if } x=2\end{array}\right. \) then \( \operatorname{Lim}_{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}= \)
(a) 0
(b) -1
(c) 1
(d) \( -1 / 2 \)
Answer: (b) -1

Question. \( \operatorname{Lt}_{x \rightarrow 0} \frac{\sqrt{\frac{1}{2}(1-\cos x)}}{x}= \)
(a) 1
(b) –1
(c) 0
(d) does not exist
Answer: (d) does not exist

Question. If \( y=\frac{1}{2} \sin ^{-1}\left(\frac{2 x y}{x^{2}+y^{2}}\right) \) and \( y < x \) then \( \lim _{y \rightarrow 0} x= \)
(a) -1
(b) 0
(c) 1
(d) \( \infty \)
Answer: (c) 1

Question. \( \operatorname{Lt}_{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2}-\cos x-\sin x}{(4 x-\pi)^{2}}= \)
(a) \( \frac{1}{16 \sqrt{2}} \)
(b) \( \frac{1}{32 \sqrt{2}} \)
(c) \( \frac{1}{16} \)
(d) \( \frac{1}{8} \)
Answer: (a) \( \frac{1}{16 \sqrt{2}} \)

Question. \( \operatorname{Lt}_{x \rightarrow 0} \frac{\operatorname{Sec} x-1}{x^{2}(\operatorname{Sec} x+1)^{2}}= \)
(a) 1/8
(b) 1/4
(c) 2
(d) 0
Answer: (a) 1/8

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

Question. \( \operatorname{Lt}_{x \rightarrow \infty} x[\log (x+1)-\log x]= \)
(a) \( e^{2} \)
(b) \( e \)
(c) 1
(d) 1/e
Answer: (c) 1

Question. \( \operatorname{Lim}_{x \rightarrow 0} \frac{27^{x}-9^{x}-3^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}}= \)
(a) 0
(b) \( 8 \sqrt{2}(\log 3)^{2} \)
(c) \( 8(\log 3)^{2} \)
(d) 1
Answer: (b) \( 8 \sqrt{2}(\log 3)^{2} \)

Question. \( \operatorname{Lt}_{x \rightarrow \infty} x\left(a^{\frac{1}{x}}-b^{\frac{1}{x}}\right)= \)
(a) 1
(b) \( \log _{e} a / b \)
(c) \( \log _{e}(a b) \)
(d) 0
Answer: (b) \( \log _{e} a / b \)

Question. The value of \( \operatorname{Lim}_{n \rightarrow \infty} \frac{1.2+2.3+3.4+\ldots+n .(n+1)}{n^{3}} \) is
(a) 1
(b) -1
(c) 1/3
(d) -1/3
Answer: (c) 1/3

Question. If \( |x|<1 \), then \( \operatorname{Lt}_{n \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2 n}\right)= \)
(a) \( \frac{1}{x} \)
(b) \( \frac{1}{1+x} \)
(c) \( \frac{1}{1-x} \)
(d) \( \frac{1}{x-1} \)
Answer: (c) \( \frac{1}{1-x} \)

Question. \( \operatorname{Lt}_{n \rightarrow \infty} \frac{2.3^{n+1}-3.5^{n+1}}{2.3^{n}+3.5^{n}}= \)
(a) 5
(b) 1/5
(c) -5
(d) 0
Answer: (c) -5

Question. \( \lim _{n \rightarrow \infty} \frac{1}{n^{4}}\left[1^{2}+\left(1^{2}+2^{2}\right)+\ldots+\left(1^{2}+2^{2}+\ldots+n^{2}\right)\right]= \)
(a) 1/6
(b) 1/16
(c) 1/12
(d) 0
Answer: (c) 1/12

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

Question. If \( a>0, \lim _{x \rightarrow \infty} \frac{[a x+b]}{x} \) is [where [.] denotes G.I.F]
(a) 0
(b) 1
(c) \( a \)
(d) \( b \)
Answer: (c) \( a \)

Question. \( \lim _{x \rightarrow 0}\left(\frac{1^{x}+2^{x}+3^{x}+\ldots+n^{x}}{n}\right)^{\frac{1}{x}}= \)
(a) \( (n !)^{n} \)
(b) \( (n !)^{1 / n} \)
(c) \( n ! \)
(d) \( \ln n ! \)
Answer: (b) \( (n !)^{1 / n} \)

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

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

Question. If \( \lim _{x \rightarrow 0}\left[1+x \ln \left(1+b^{2}\right)\right]^{\frac{1}{x}}=2 b \sin ^{2} \theta, b>0 \) and \( \theta \in(-\pi, \pi) \) then the value of \( \theta \) is
(a) \( \pm \frac{\pi}{6} \)
(b) \( \pm \frac{\pi}{3} \)
(c) \( \pm \frac{\pi}{8} \)
(d) \( \pm \frac{\pi}{2} \)
Answer: (d) \( \pm \frac{\pi}{2} \)

Question. \( \lim _{x \rightarrow 1}\left(\log _{3} 3 x\right)^{\log _{x} 3}= \)
(a) \( e^{-1} \)
(b) \( e \)
(c) -1
(d) 1
Answer: (b) \( e \)

Question. If \( p \) and \( q \) are the roots of the quadratic equation \( a x^{2}+b x+c=0 \) then \( \lim _{x \rightarrow p}\left(1+a x^{2}+b x+c\right)^{\frac{1}{x-p}}= \)
(a) \( a(p-q) \)
(b) \( \log [a(p-q)] \)
(c) \( e^{a(p-q)} \)
(d) \( e^{a(q-p)} \)
Answer: (c) \( e^{a(p-q)} \)

Question. \( \operatorname{Lt}_{x \rightarrow \infty}\left(\frac{x^{2}+5 x+3}{x^{2}+x+2}\right)^{x}= \)
(a) \( e^{4} \)
(b) \( e^{3} \)
(c) \( e^{2} \)
(d) \( 2^{4} \)
Answer: (a) \( e^{4} \)

Question. \( \operatorname{Lim}_{x \rightarrow \infty}\left\{\left(\frac{x+a}{x+1}\right)^{x}+\sin \frac{1}{x}\right\} \) is equal to
(a) \( e^{a-1} \)
(b) \( e^{1-a} \)
(c) \( e \)
(d) 0
Answer: (b) \( e^{1-a} \)

Question. \( \lim _{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{\frac{1}{x}}= \)
(a) \( e \)
(b) \( e^{2} \)
(c) \( e^{-1} \)
(d) \( e^{-2} \)
Answer: (b) \( e^{2} \)

Question. \( \lim _{x \rightarrow \infty}\left(1+\frac{\lambda}{x}+\frac{\mu}{x^{2}}\right)^{2 x}=e^{4} \) then \( \lambda= \) ( \( \mu \in \mathbf{R} \) )
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. A function \( f: R \rightarrow R \) is such that \( f(1)=3 \) and \( f^{\prime}(1)=6 \). Then \( \lim _{x \rightarrow 0}\left[\frac{f(1+x)}{f(1)}\right]^{1 / x}= \)
(a) 1
(b) \( e^{2} \)
(c) \( e^{1 / 2} \)
(d) \( e^{3} \)
Answer: (b) \( e^{2} \)

Question. The integer n for which \( \lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}} \) is a finite non-zero number is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. The values of a, b, and c such that \( \lim _{x \rightarrow 0} \frac{a x e^{x}-b \log (1+x)+c x e^{-x}}{x^{2} \sin x}=2 \)
(a) \( a=3, b=12, c=9 \)
(b) \( a=1, b=2, c=4 \)
(c) \( a=2, b=10, c=8 \)
(d) \( a=3, b=-12, c=-9 \)
Answer: (a) \( a=3, b=12, c=9 \)

Question. \( \lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\sin x}{x^{3}} \)
(a) 1/2
(b) 1/3
(c) 1/4
(d) 1/5
Answer: (b) 1/3

Question. \( \lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e\left(1-\frac{x}{2}\right)}{(1-\cos x)} \)
(a) \( \frac{1}{2} e \)
(b) \( \frac{1}{4} e \)
(c) \( \frac{11}{12} e \)
(d) \( \frac{1}{12} e \)
Answer: (c) \( \frac{11}{12} e \)

Question. If [x] denotes the greatest integer less than or equal to x then \( \operatorname{Lim}_{n \rightarrow \infty} \frac{[x]+[2 x]+\ldots+[n x]}{n^{2}}= \)
(a) x/2
(b) x/3
(c) x
(d) 0
Answer: (a) x/2

Question. \( \lim _{n \rightarrow \infty} \frac{\{x\}+\{2 x\}+\ldots+\{n x\}}{n^{2}} \)
(a) 1/2
(b) 0
(c) -1
(d) 2
Answer: (b) 0

Question. \( \operatorname{Lt}_{x \rightarrow 0}\left[\frac{\sin |x|}{|x|}\right] \) Where [.] denotes the greatest integer function.
(a) 0
(b) 1
(c) -1
(d) does not exist
Answer: (a) 0

Question. Let \( f: R \rightarrow R \) be a positive increasing function with \( \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 \), then \( \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}= \)
(a) 2/3
(b) 3/2
(c) 3
(d) 1
Answer: (d) 1