CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 4 - MCQExams.com


$$\displaystyle \lim_{x\rightarrow 0}\displaystyle \frac{(1+a^{3})+8e^{1/x}}{1+(1-b^{3})e^{1/x}}=2$$ then
  • $$a=1,b=(-3)^{1/3}$$
  • $$a=1,b=3^{1/3}$$
  • $$a=-1,b=(-3)^{1/3}$$
  • $$a=1,b=0$$
If $$f(x)=(x)^{\tfrac{1}{x-1}}$$ for $$x\neq 1$$ and $$\mathrm{f}$$ is continuous at $$\mathrm{x}=1$$ then $$\mathrm{f}(1)=$$
  • $$\mathrm{e}$$
  • $$\mathrm{e}^{-1}$$
  • $$\mathrm{e}^{-2}$$
  • $$\mathrm{e}^{2}$$
Assertion (A): $$f(x)=\displaystyle \frac{\sin\{[x]\pi\}}{1+x^{2}}$$ is continuous on $$\mathrm{R}$$ (where $$[x]$$ denotes greatest integer function of $$x $$).
Reason (R): Every constant function is continuous on $$\mathrm{R}$$
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true and R is not the correct explanation of A
  • A is true but R is false
  • R is true but A is false
The values of $$p$$ and $$q$$ for which the function $$\mathrm{f}(\mathrm{x}) = \left\{\begin{array}{ll}
\dfrac{\sin(\mathrm{p}+1)\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}} & , \mathrm{x}<0\\
\mathrm{q} & , \mathrm{x}=0\\
\dfrac{\sqrt{\mathrm{x}+\mathrm{x}^{2}}-\sqrt{\mathrm{x}}}{\mathrm{x}^{3/2}} & , \mathrm{x}>0
\end{array}\right.$$
is continuous for all $$\mathrm{x}$$ in $$\mathrm{R}$$, are
  • $$\displaystyle \mathrm{p}=\dfrac{1}{2},\ \displaystyle \mathrm{q}=-\dfrac{3}{2}$$
  • $$\displaystyle \mathrm{p}=\dfrac{5}{2},\mathrm{q}=\dfrac{1}{2}$$
  • $$\displaystyle \mathrm{p}=-\dfrac{3}{2},\ \displaystyle \mathrm{q}=\dfrac{1}{2}$$
  • $$\displaystyle \mathrm{p}=\dfrac{1}{2},\mathrm{q}=\dfrac{3}{2}$$
If $$\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} x+\lambda , & -1<x<3 \end{matrix} \\ \begin{matrix} 4, & x=3 \end{matrix} \\ \begin{matrix} 3x-5, & x>3 \end{matrix} \end{cases}$$ is continuous at $$x=3$$ then tha value of $$\lambda$$ is
  • $$-1$$
  • $$0$$
  • $$1$$
  • does not exits

The function $$\displaystyle \mathrm{f}({x})=\begin{cases}\dfrac{1-\sin x}{(\pi-2x)^{2}} & x  \neq\dfrac{\pi}{2}\\ \mathrm{k}& {x}=\dfrac{\pi}{2}\end{cases}$$ is continuous at $$\displaystyle {x}=\dfrac{\pi}{2}$$ then $$\mathrm{k}$$ is equal to
  • $$\dfrac{1}{8}$$
  • 4
  • 3
  • 1
The value of $$f(0)$$ so that the function $$\mathrm{f}({x})$$$$=\displaystyle \frac{1-\cos(1-\cos x)}{x^{4}}$$ is continuous everywhere is
  • $$\displaystyle \frac{1}{8}$$
  • $$\displaystyle \frac{1}{2}$$
  • $$\displaystyle \frac{1}{4}$$
  • $$\displaystyle \frac{1}{3}$$
$$\underset{x\rightarrow0}{lim}\displaystyle\frac{1-cos^{3}x+sin^{3}x+\ell n(1+x^{3})+\ell n(1+cos\,\,x)}{x^{2}-1+2\,cos^{2}x+tan^{4}x+sin^{3}x}$$ is equal to -
  • $$\,\,\displaystyle\frac{3}{4}$$
  • $$\,\,ln2$$
  • $$\,\,\displaystyle\frac{ln2}{4}$$
  • $$\,\,3/2$$
The value of $$p$$ for which the function

$$f(x)=\displaystyle \left\{ \begin{array}{rl} \dfrac { (4^{ { x } }-1)^{ 3 } }{ \sin\dfrac { { x } }{ { p } } \log(1+\dfrac { { x }^{ 2 } }{ 3 } ) }  & { ;\quad x\neq 0 } \\ 12(\log  4)^{ 3 } & { ;\quad x=0\quad  } \end{array} \right. $$ is continuous at $${x}=0$$, is
  • $$4$$
  • $$2$$
  • $$3$$
  • $$1$$
If $$f(x)=\displaystyle \frac { 1 }{ x(3x+1) } $$ then at $${x}=0,\mathrm{f}({x})$$ is
  • continuous
  • discontinuous
  • not determined
  • $$\displaystyle \lim_{x\rightarrow 0 }f(x)=2$$
Assertion(A):
$$f(x)=x(\displaystyle \frac{1+e^{1/x}}{1-e^{1/x}})(x\neq 0)$$ , $${f}(0)=0$$ is continuous at $${x}=0$$.
Reason(R) A function is said to be continuous at $$a$$ if both limits are exists and equal to $$\mathrm{f}({a})$$ .
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true and R is not the correct explanation of A
  • A is true but R is false
  • R is true but A is false

The value of $$\mathrm{f}({0})$$ so that the function $$\displaystyle \mathrm{f}({x})=\frac{\sqrt{1+x}-\sqrt[3]{1+x}}{x}$$ becomes continuous is equal to
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{4}$$
  • $$2$$
  • $$\dfrac{1}{3}$$
If $$\displaystyle f(x) = \sqrt {\frac{{x - \sin x}}{{x + {{\cos }^2}x}}} $$ then $$\mathop {\lim }\limits_{x \to \infty } f(x)$$  is
  • O
  • $$\infty$$
  • 1
  • None of these
$$\displaystyle\lim_{x \rightarrow \infty}(1^x + 2^x + 3^x+.........+n^x)^{1/x}$$ is
  • $$\ln (n!)$$
  • $$n$$
  • $$n!$$
  • $$0$$
$$\displaystyle \lim_{x\to\infty}{\displaystyle \frac{(2x + 1)^{40}(4x - 1)^5}{(2x + 3)^{45}}}$$ is equal to
  • 16
  • 24
  • 32
  • 8
If $$f(x) =\left\{\begin{matrix}
\dfrac{8^{x}-4^{x}-2^{x}+1^{x}}{x^{2}},x>0 & \\
e^{x}\sin x+\pi x+\lambda \ln 4,x\leq 0 &
\end{matrix}\right.$$is continuous at $$x= 0$$, then $$\lambda$$ is a
  • rational number
  • irrational number
  • natural number
  • complex number
$$\displaystyle \lim_{n\to\infty}{\displaystyle \frac{n(2n + 1)^2}{(n + 2)(n^2 + 3n - 1)}}$$ is equal to
  • $$0$$
  • $$2$$
  • $$4$$
  • $$\infty$$
Let $$\mathrm{f}:\mathrm{R}\rightarrow \mathrm{R}$$ be defined by $$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}
\mathrm{k}-2\mathrm{x}, \mathrm{i}\mathrm{f}   \mathrm{x}\leq-1\\
2\mathrm{x}+3, \mathrm{i}\mathrm{f}   \mathrm{x}>-1
\end{array}\right.$$ be continous. then find possible value of $$\mathrm{k}$$ is
  • $$0$$
  • $$\displaystyle -\frac{1}{2}$$
  • $$-1$$
  • $$1$$
$$\lim_{n\rightarrow \infty }\left ( \frac{F\left ( n \right )}{G\left ( n \right )} \right )^{n}-\lim_{n\rightarrow \infty }\left ( \frac{f\left ( n \right )}{g\left ( n \right )} \right )^{n}=$$
  • $$e^{3/2}-e$$
  • $$e^{-3/2}-e^{-1}$$
  • $$e^{3/2}$$
  • $$e^{-1}$$
If $$f(x) = \displaystyle \left\{\begin{matrix}x - 1, & x \geq 1 \\ 2x^2 - 2, & x < 1\end{matrix}\right. , g(x) = \left\{\begin{matrix}x + 1, & x > 0 \\ -x^2 + 1, & x \leq 0\end{matrix}\right.$$, and $$h(x) = |x|$$, then $$\displaystyle \lim_{x \rightarrow 0} f(g (h (x)))$$ is
  • 0
  • 1
  • 2
  • 3
If $$\displaystyle \lim_{x\to0}{\displaystyle \frac{x^n - \sin^nx}{x - \sin^nx}}$$ is non-zero finite, then $$n$$ must be equal
  • 4
  • 1
  • 2
  • 3
$$\displaystyle \lim_{x\to1}{\displaystyle \frac{1-x^2}{\sin 2\pi x}}$$ is equal to
  • $$\displaystyle \frac{1}{2\pi}$$
  • $$\displaystyle \frac{-1}{\pi}$$
  • $$\displaystyle \frac{-2}{\pi}$$
  • None of these
$$\displaystyle \lim_{n \rightarrow \infty} \frac{-3n + (-1)^n}{4n - (-1)^n}$$ is equal to
  • $$\displaystyle \frac{3}{4}$$
  • $$0$$ if $$n$$ is even
  • $$\displaystyle- \frac{3}{4}$$
  • none of these
If $$\displaystyle \lim _{ x\to\infty  }\left\{\displaystyle \frac{x^3 + 1}{x^2 +1} - (ax + b)  \right\}  = 2$$, then
  • $$ a = 1, b = 1$$
  • $$ a = 1, b = 2$$
  • $$ a = 1, b = -2$$
  • None of these
If $$\displaystyle f(x) = \left\{\begin{matrix}x^2+2, & x \geq 2\\ 1-x, & x < 2\end{matrix}\right.$$ and $$g(x) = \left\{\begin{matrix}2x, & x > 1\\ 3-x, & x \leq 1\end{matrix}\right.$$, then the value of $$\displaystyle \lim_{x \rightarrow 1} f(g(x))$$ is ............
  • $$6$$
  • $$4$$
  • $$8$$
  • $$2$$
$$\displaystyle \lim_{x\to2} \left( \left( \displaystyle \frac{x^3 - 4x}{x^3 - 8} \right)^{-1} - \left( \displaystyle \frac{x + \sqrt{2x}}{x - 2} - \displaystyle \frac{\sqrt {2}}{\sqrt{x} - \sqrt{2}} \right)^{-1} \right)$$ is equal to
  • $$\dfrac{1}{2}$$
  • $$2$$
  • $$1$$
  • None of these
$$\displaystyle f(x) = \frac{3x^2 + ax + a + 1}{x^2 + x - 2} $$ and $$\displaystyle \lim_{x \rightarrow - 2} f(x)$$ exists. 
Then the value of $$(a- 4)$$ is?
  • $$9$$
  • $$10$$
  • $$11$$
  • $$12$$
$$\displaystyle \lim_{n \rightarrow \infty} {^{n}C_{c}}\left(\dfrac {m}{n}\right)^{x}\left(1-\dfrac {m}{n}\right)^{n-x}$$ equal to
  • $$\dfrac {M^{x}}{x\ !}.e^{-m}$$
  • $$\dfrac {M^{x}}{x\ !}.e^{m}$$
  • $$0$$
  • $$\dfrac {m^{x+1}}{me^{m}x\ !}$$
$$\displaystyle \lim_{x\to0}\left( x^{-3}\sin{3x} + ax^{-2} + b \right)$$ exists and is equal to 0, then
  • $$a = -3$$ and $$b = \dfrac{9}{2}$$
  • $$a = 3$$ and $$b = \dfrac{9}{2}$$
  • $$a = -3$$ and $$b = -\dfrac{9}{2}$$
  • $$a = 3$$ and $$b = -\dfrac{9}{2}$$
$$\displaystyle \lim _{ x\to\infty } \left( \frac { x^{ 2 }+2x-1 }{ 2x^2-3x-2 }  \right) ^{\LARGE  \frac { 2x+1 }{ 2x-1 }  }$$ is equal to
  • $$0$$
  • $$\infty$$
  • $$\dfrac{1}{2}$$
  • None of these
If $$p\left( x \right) ={ a }_{ 0 }+{ a }_{ 1 }x+...+{ a }_{ n }{ x }^{ n }$$ and $$\left| p\left( x \right)  \right| \le \left| { e }^{ x-1 }-1 \right| $$ for all $$x\ge 0,$$ then $$\left| { a }_{ 1 }+2{ a }_{ 2 }+3{ a }_{ 3 }+...+n{ a }_{ n } \right| $$
  • $$\le 1$$
  • $$\ge 1$$
  • $$\ge 0$$
  • $$\le 0$$
The value of $$\displaystyle\lim_{x\rightarrow\infty}{\frac{\cot^{-1}{(x^{-a}\log_a{x})}}{\sec^{-1}{a^x\log_x{a}}}}$$ for $$(a>1)$$ is equal to?
  • $$1$$
  • $$0$$
  • $$\displaystyle\frac{\pi}{2}$$
  • Does not exist
The value of 
$$\displaystyle \lim_{x \rightarrow \pi/6} \frac{2 \sin^2 x + \sin  x-1}{2 \sin^2 x - 3  \sin  x + 1} $$
  • $$3$$
  • $$-3$$
  • $$6$$
  • $$0$$
Let $$f(x)=\sin x$$, $$g(x)=\left [ x+1 \right ]$$ and $$g(f(x))=h(x)$$, where [.] is the greatest integer function. Then $$h^+\left ( \displaystyle \dfrac{\pi }{2} \right )$$ is
  • non existent
  • $$1$$
  • $$-1$$
  • none of these
Which one of the following statement is true?
  • if $$\displaystyle \lim_{x\to c}f(x).g(x)$$ and $$\displaystyle \lim_{x\to c}f(x)$$ exist, then $$\displaystyle \lim_{x\to c}g(x)$$ exists
  • if $$\displaystyle \lim_{x\to c}f(x).g(x)$$ exists, then $$\displaystyle \lim_{x\to c}f(x)$$ and $$\displaystyle \lim_{x\to c}g(x)$$ exist
  • if $$\displaystyle \lim_{x\to c}(f(x)+g(x))$$ and $$\displaystyle \lim_{x\to c}f(x)$$ exist, then $$\displaystyle \lim_{x\to c}g(x)$$ exists
  • if $$\displaystyle \lim_{x\to c}(f(x)+g(x))$$ exists, then $$\displaystyle \lim_{x\to c}f(x)$$ and $$\displaystyle \lim_{x\to c}g(x)$$ exists
$$\displaystyle \lim_{n\to\infty }\frac{n^{p}\sin ^{2}\left ( n! \right )}{n+1}$$, $$0<p<1$$, is equal to
  • $$0$$
  • $$\infty $$
  • $$1$$
  • $$none\ of\ these$$
$$\displaystyle \lim_{x \rightarrow \infty} (\sqrt{x^2 + 8x + 3} - \sqrt{x^2 + 4x + 3}) =$$
  • $$0$$
  • $$\infty$$
  • $$2$$
  • $$\dfrac{1}{2}$$
Let $$f\left ( x \right )=\begin{cases}\sin x, x\neq n\pi 
                   \\ 2,  x=n\pi \end{cases}$$, where $$n\epsilon \mathbb{Z}$$ and
$$g\left ( x \right )=\begin{cases}x^{2}+1, x\neq 2 \\
              3, x=2 \end{cases}$$.
Then $$\displaystyle \lim_{x\to 0}g\left ( f\left ( x \right ) \right )$$ is
  • $$0$$
  • $$1$$
  • $$3$$
  • none of these
$$f\left( x \right)=\begin{cases} \sin { x } \qquad ;\qquad x\neq n\pi ,n=0,\pm 1,\pm 2,\pm 3..... \\ 2\qquad \qquad ;\qquad otherwise \end{cases}$$ and $$g\left( x \right) =\begin{cases} { x }^{ 2 }+1\qquad ;\qquad x\neq 0 \\ 4\qquad \qquad ;\qquad x=0 \end{cases}.$$ 
Then $$\lim _{ x\rightarrow 0 }{ g\left( f\left( x \right)\right)} $$ is
  • $$1$$
  • $$4$$
  • $$5$$
  • non-existent
$$\displaystyle \lim_{x\rightarrow \dfrac{\pi}{2}}\dfrac{\left ( 1-\tan \dfrac{x}{2} \right )\left ( 1-\sin x \right )}{\left ( 1+\tan \dfrac{x}{2} \right )\left ( \pi -2x \right )^{3}}$$ is
  • $$0$$
  • $$\displaystyle \frac{1}{32}$$
  • $$\infty $$
  • $$\displaystyle \frac{1}{8}$$
Which one of the following statements is true?
  • If $$\displaystyle\lim_{x\rightarrow c}{f(x).g(x)}$$ and $$\displaystyle\lim_{x\rightarrow c}{f(x)}$$ exist, then $$\displaystyle\lim_{x\rightarrow c}{g(x)}$$ exists.
  • If $$\displaystyle\lim_{x\rightarrow c}{f(x).g(x)}$$ exists, then $$\displaystyle\lim_{x\rightarrow c}{f(x)}$$ and $$\displaystyle\lim_{x\rightarrow c}{g(x)}$$ exist.
  • If $$\displaystyle\lim_{x\rightarrow c}{f(x)+g(x)}$$ and $$\displaystyle\lim_{x\rightarrow c}{f(x)}$$ exist, then $$\displaystyle\lim_{x\rightarrow c}{g(x)}$$ also exists.
  • If $$\displaystyle\lim_{x\rightarrow c}{f(x)+g(x)}$$ exists, then $$\displaystyle\lim_{x\rightarrow c}{f(x)}$$ and $$\displaystyle\lim_{x\rightarrow c}{g(x)}$$ also exist.
If $$\displaystyle \lim_{x\rightarrow 0}(f(x)\:g(x))$$ exists for any functions $$f$$ and $$g$$ then
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ and $$\displaystyle \lim_{x\rightarrow a}g(x)$$ exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ exist but $$\displaystyle \lim_{x\rightarrow a}g(x)$$ may not exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ may not exist but $$\displaystyle \lim_{x\rightarrow a}g(x)$$ exist
  • $$\displaystyle \lim_{x\rightarrow a}f(x)$$ and $$\displaystyle \lim_{x\rightarrow a}g(x)$$ may not exist
Evaluate $$\displaystyle \lim_{n\rightarrow \infty }\left [ \frac{n!}{n^{n}} \right ]^{1/n}$$.
  • $$\displaystyle\frac{1}{e}$$
  • $$\displaystyle\frac{1}{t}$$
  • $$\displaystyle\frac{1}{n}$$
  • $$none\ of\ above$$
$$\displaystyle \lim_{x\rightarrow\infty}\left(\frac{\sqrt{(1 - \cos x)+ \sqrt{(1 - \cos x)+ \sqrt(1 - \cos x)+...\infty) - 1}}}{x^2}\right)$$ equals to
  • 0
  • $$\displaystyle\frac{1}{2}$$
  • 1
  • 2
State whether the given statement is true or false 
If f(x) satisfies the relation, f(x+y)=f(x)+f(y) for all $$x, y\epsilon R$$ and f(1)=5, then find $$\sum_{n=1}^{m}f\left ( n \right )$$. 
  • True
  • False
If [.] denotes, GIF , then $$\underset{x \rightarrow 0}{lt} \left( \left[\dfrac{2018 sin^{-1} x}{x}\right] + \left[\dfrac{2020x}{tan^{-1} x}\right]\right)$$ = 
  • 4038
  • 4037
  • 4036
  • 4039
The value of $$\displaystyle \lim_{x \rightarrow 0} \left(\dfrac{\sin x}{x}\right)^{1/x^{2}}$$ is 
  • $$e^{-1/6}$$
  • $$e^{1/6}$$
  • $$e^{-1/3}$$
  • $$e^{1/3}$$
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is incorrect but Reason is correct 
let a, b, c are non zero constant number then $$\lim_{r\rightarrow\infty}\displaystyle\frac{cos\displaystyle\frac{a}{r}-cos\displaystyle\frac{b}{r}cos\displaystyle\frac{c}{r}}{sin\displaystyle\frac{b}{r}sin\displaystyle\frac{c}{r}}$$ equals to
  • $$\displaystyle\frac{a^2 + b^2 - c^2}{2bc}$$
  • $$\displaystyle\frac{c^2 + a^2 - b^2}{2bc}$$
  • $$\displaystyle\frac{b^2 + c^2 - a^2}{2bc}$$
  • independent of a, b and c
Evaluate $$\displaystyle \lim_{n\rightarrow \infty }\left [ \left ( 1+\frac{1}{n^{2}} \right )\left ( 1+\frac{2^{2}}{n^{2}} \right )\left ( 1+\frac{3^{2}}{n^{2}} \right )......\left ( 1+\frac{n^{2}}{n^{2}} \right ) \right ]^{1/n}$$
  • $$2e ^\left(\dfrac{\pi - 4}{2}\right)$$
  • $$2e ^\left(\dfrac{\pi - 2}{2}\right)$$
  • $$2e^ \left(\dfrac{\pi - 4}{4}\right)$$
  • $$2e^ \left(\dfrac{\pi - 4}{3}\right)$$
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