CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 3 - MCQExams.com

lf $$f(x)=\displaystyle \begin{cases}\dfrac{a^{2[x]+\{x\}}-1}{2[x]+\{x\}};x\neq 0 \\ \log a;x=0 \end{cases}$$ where $$[.\ ]$$ and $$\{.\ \}$$ denote integral and fractional part respectively, then
  • $$f(x)$$ is continuous at $$x=0$$
  • $$f(x)$$ is discontinuous at $$x=0$$
  • $$f(x)$$ is continuous $$\forall x\in R$$
  • $$f(x)$$ is differentiable at $$x=0$$

 lf $$\displaystyle { f }({ x })=\sqrt { \frac { { x }-\sin ^{ 2 }{ x }  }{ { x }+\cos { x }  }  } $$,then $$\displaystyle \lim _{ x\rightarrow \infty  } f(x)$$=
  • $$\dfrac{1}{2}$$
  • $$-1$$
  • $$0$$
  • $$1$$
The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ \frac { \sin { \left( \pi \cos ^{ 2 }{ x }  \right)  }  }{ { x }^{ 2 } }  } $$ is
  • $$-\pi$$
  • $$\displaystyle \frac { \pi  }{ 2 } $$
  • $$\pi$$
  • $$\displaystyle \frac { 3\pi  }{ 2 } $$
Let $$f(x)=\cos2x.\cot\left (\displaystyle  \frac{\pi }{4}-x \right )$$ If $$f$$ is continuous at $$x=\displaystyle \frac{\pi}{4}$$ then the value of $$f(\displaystyle \frac{\pi}{4})$$ is equal to
  • $$2$$
  • $$-2$$
  • $$\displaystyle \frac{-1}{2}$$
  • $$\displaystyle \frac{1}{2}$$
$$\displaystyle \lim_{x\rightarrow \infty }x\displaystyle \cos\left(\frac{\pi}{8x}\right)\sin\left(\frac{\pi}{8x}\right)=$$
  • $$\displaystyle \pi$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\dfrac{\pi}{8}$$
  • $$\displaystyle \frac{\pi}{4}$$

$$\displaystyle \lim_{x\rightarrow \infty }(\sin\sqrt{x+1}-\sin\sqrt{x})=$$
  • 2
  • -2
  • 0
  • None of these

 $$\displaystyle \lim_{x\rightarrow\infty}\frac{\sin^{4}x-\sin^{2}x+1}{\cos^{4}x-\cos^{2}x+1}$$ is equal to
  • $$0$$
  • $$1$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
$$f(x)=\left\{\begin{matrix}[x]+[-x], & \\  \lambda ,& \end{matrix}\right.\begin{matrix}x\neq 2 & \\  x=2& \end{matrix},$$ then f(x) is continuous at $$x=2$$ provided $$\lambda $$ is:
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$

$$\displaystyle \lim_{x\rightarrow \infty }2^{-x}\sin(2^{x})$$
  • $$1$$
  • $$0$$
  • $$2$$
  • does not exist
Let $$f : R\rightarrow R$$ be any function, Define
$$g:R\rightarrow R$$ by $$g(x)=|f(x)|\forall x$$, then
  • $$g$$ is continuous if $$f$$ is not continuous
  • $$g$$ is not continuous if $$f$$ is not continuous
  • $$g$$ is continuous if $$f$$ is continuous
  • $$g$$ is differentiable if $$f$$ is differentiable
$$\displaystyle \lim_{x\rightarrow \infty }\frac{2x+7\sin x}{4x+3\cos x}=$$
  • $$1$$
  • $$-1$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
The function $$y=3\sqrt{x}-|x-1|$$ is continuous at
  • $$x<0$$
  • $$x\geq 1$$
  • $$0\leq x\leq 1$$
  • $$x\geq 0$$

 lf $$\mathrm{f}(\mathrm{x})=\left\{\begin{matrix}1+x &x\leq 1 \\  3-ax^{2}& x>1\end{matrix}\right.$$ is continuous at $${x}=1$$ then $${a}=({a}>0)$$
  • $$1$$
  • $$2$$
  • $$0$$
  • $$3$$

$$\displaystyle Lt_{x\rightarrow 0^+}(sinx)^{\tan x}=$$
  • $${e}$$
  • $$e^{2}$$
  • $$-1$$
  • 1
The function $$\displaystyle \mathrm{f}({x})=\frac{1+\sin x-\cos x}{1-\sin x-\cos x}$$ is not defined at $${x}=0$$. The value of $$\mathrm{f}(\mathrm{0})$$ so that $$\mathrm{f}({x})$$ is continuous at $${x}=0$$ is
  • $$1$$
  • $$-1$$
  • $$0$$
  • $$2$$
lf the function $$\displaystyle \mathrm{f}({x})=\frac{e^{x^{2}}-\cos {x}}{x^{2}}$$ for $$x \neq 0$$ is continuous at $${x}=0$$ then $$\mathrm{f}(\mathrm{0})=$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{2}$$
  • $$2$$
  • $$\dfrac{1}{3}$$

$$\displaystyle \lim_{\mathrm{x}\rightarrow \pi }(1- 4 \tan \mathrm{x} )^{\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{x}}=$$
  • $$\mathrm{e}$$
  • $$\mathrm{e}^{4}$$
  • $$\mathrm{e}^{-1}$$
  • $$\mathrm{e}^{-4}$$
$$\displaystyle \lim_{n\rightarrow \infty }(\pi n)^{2/n}=$$
  • 0
  • 1
  • 2
  • 3
$$\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} \displaystyle\frac { { x }^{ 5 }-32 }{ x-2 } , & x\neq 2 \end{matrix} \\ \begin{matrix} k, & x=2 \end{matrix} \end{cases}$$ is continuous at $$x=2$$, then the value of $$k$$ is 
  • $$10$$
  • $$15$$
  • $$35$$
  • $$80$$
If $$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\mathrm{a}^{2}\cos^{2}\mathrm{x}+\mathrm{b}^{2}\sin^{2}\mathrm{x},\mathrm{x}\leq 0\\\mathrm{e}^{\mathrm{a}\mathrm{x}+\mathrm{b}},\mathrm{x}>0\end{array}\right.$$ is continuous at $$\mathrm{x}=0$$ then
  • $$2\log|\mathrm{a}|=\mathrm{b}$$
  • $$2\log|\mathrm{b}|=\mathrm{e}$$
  • $$\log a=2\log|\mathrm{b}|$$
  • $$\mathrm{a}=\mathrm{b}$$
$$\displaystyle \lim_{n\rightarrow \infty }\left(\displaystyle \frac{e^{n}}{\pi}\right)^{1/n}=$$
  • 0
  • 1
  • $$\mathrm{e}^{2}$$
  • $$\mathrm{e}$$
$$\displaystyle \lim_{x\rightarrow 1}(2-x)^{\displaystyle \tan( \frac{\pi x}{2})}=$$
  • $$e^{\displaystyle \frac{1}{\pi}}$$
  • $$e^{\displaystyle \frac{2}{\pi}}$$
  • $$-e^{\displaystyle \frac{2}{\pi}}$$
  • $$\mathrm{e}$$
lf the function defined by $$\mathrm{f}({x})=\displaystyle \frac{\sin 3(x-p)}{\sin 2(x-p)}$$ for $${x}\neq {p}$$ is continuous at $${x}={p}$$ then $$\mathrm{f}({p})=$$
  • $$\dfrac{3}{2}$$
  • $$\dfrac{2}{3}$$
  • $$6$$
  • $$\dfrac{1}{6}$$

If the function $$\displaystyle \mathrm{f}({x})=\begin{cases}\dfrac{2^{x+2}-16}{4^{x}-16}&&  for {x}\neq 2\\ \mathrm{A} && x =2\end{cases}$$ is continuous at $$x =2$$, then $$\mathrm{A}=$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{4}$$
  • $$0$$

 $$f(x)=x\left[3-\displaystyle \log\left(\frac{\sin {x}}{x}\right)\right]-2$$ to be continuous at $${x}=0$$, then $$\mathrm{f}({0})=$$
  • $$0$$
  • $$2$$
  • $$-2$$
  • $$3$$

 Let $$\displaystyle \mathrm{f}({x})=\begin{cases} \dfrac{(e^{kx}-1).\sin kx}{x^{2}} & for \ {x}\neq 0   \\ 4 & for \ {x} =0\end{cases}$$ is continuous at $${x}=0$$ then $${k}=$$
  • $$\pm 1$$
  • $$\pm 2$$
  • 0
  • $$\pm 3$$

 lf $$f(x)=
\left\{\begin{matrix} (1+|\sin x|)^{\displaystyle \frac{a}{|\sin x|}}&-\displaystyle \frac{\pi}{6}<x<0\\  b&x=0 \\ e^{\displaystyle \frac{\tan 2x}{\tan 3x}} &0<x<\displaystyle \frac{\pi}{6}\end{matrix}\right.$$ is

continuous at $$\mathrm{x}=0$$ then
  • $$a=e^{2/3},b=\dfrac{2}{3}$$
  • $$a=\dfrac{2}{3},b=e^{2/3}$$
  • $$a=\dfrac{1}{3},b=e^{1/3}$$
  • $$a=e^{1/3},b=e^{1/3}$$
lf the function $$f(x)=\begin{cases}\dfrac{k\cos x}{\pi-2x}, & x\neq\dfrac{\pi}{2}\\ 3 & at x=\dfrac{\pi}{2}\end{cases}$$is continuous at $$\displaystyle {x}=\dfrac{\pi}{2}$$ then $${k}=$$
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
The function $$f(x)=\begin{cases} 0,&  \text{x  is irrational }\\  1,& \text{x is rational }\end{cases}$$ is
  • continuous at $$x=1$$
  • discontinuous only at $$0$$
  • discontinuous only at 0,1
  • discontinuous everywhere

The function $$\displaystyle \mathrm{f}(\mathrm{x})=\frac{\cos x-\sin x}{\cos 2x}$$ is not defined at $$x=\displaystyle \frac{\pi}{4}$$ The value of $$f\left(\displaystyle \frac{\pi}{4}\right)$$ so that $$\mathrm{f}(\mathrm{x})$$ is continuous at $$x=\displaystyle \frac{\pi}{4}$$ is
  • $$\displaystyle \frac{1}{\sqrt{2}}$$
  • $$\sqrt{2}$$
  • $$-\sqrt{2}$$
  • $$1$$
The value of $$f(0)$$ so that the function
$$f(x)=\dfrac{\displaystyle \log\left(1+\dfrac{x}{a}\right)-\log\left(\begin{array}{l}1-\dfrac{x}{b}\end{array}\right)}{x}, (x\neq 0)$$ is continuous at $$x = 0$$ is :
  • $$\displaystyle \frac{a+b}{ab}$$
  • $$\displaystyle \frac{a-b}{ab}$$
  • $$\displaystyle \frac{ab}{a+b}$$
  • $$\displaystyle \frac{ab}{a-b}$$
The value of $$\mathrm{f}(\mathrm{0})$$ for the function $$\mathrm{f}({x})=\displaystyle \frac{2-\sqrt{(x+4)}}{\sin 2x}, x\ne 0$$ is continuous at $${x}=0$$ is
  • $$\displaystyle \frac{1}{8}$$
  • $$\displaystyle \frac{1}{4}$$
  • $$\displaystyle \frac{-1}{8}$$
  • $$-\displaystyle \frac{1}{4}$$
lf $$f(x)=\left\{\begin{array}{l}\dfrac{1-\sqrt{2}\sin x}{\pi-4x}  x\neq\frac{\pi}{4}\\a,x=\frac{\pi}{4}\end{array}\right.$$ is continuous at $$x=\displaystyle \frac{\pi}{4}$$ then $$a=$$
  • $$4$$
  • $$2$$
  • $$1$$
  • $$\dfrac{1}{4}$$
If $$ \displaystyle f(x)=\left\{\begin{array}{ll}\dfrac{\sqrt{1+kx}-\sqrt{1-x}}{x} & \mathrm{f}\mathrm{o}\mathrm{r}-\mathrm{l} \leq x<0\\2x^{2}+3x-2 & \mathrm{f}\mathrm{o}\mathrm{r} 0\leq x\leq 1\end{array}\right.$$ is continuous at $$x = 0$$ then $$k$$ is:
  • $$-4$$
  • $$-3$$
  • $$-5$$
  • $$-1$$
$$f(x)=\begin{cases}\dfrac{x^{3}+x^{2}-16x+20}{(x-2)^{2}} & if\  x\neq 2\\ k & if\  x=2\end{cases}$$
$$\mathrm{f}({x})$$ is continuous at $${x}=2$$ then $$f(2)=$$
  • $${k}=3$$
  • $${k}=5$$
  • $${k}=7$$
  • $${k}=9$$
$$f(x)=\dfrac{p+q^{\frac{1}{x}}}{r+s^{\frac{1}{x}}},
s<1, q<1,r\neq 0, \mathrm{f}(\mathrm{0})=1$$, is left continuous at $$x =0$$ then
  • $${p}=0$$
  • $${p}={r}$$
  • $${p}={q}$$
  • $$p\neq q$$
lf $$f$$ : $$R\rightarrow R$$ is defined by $$f(x)=\left\{\begin{array}{ll}\displaystyle \frac{\cos 3x-\cos x}{x^{2}} & \mathrm{f}\mathrm{o}\mathrm{r} x\neq 0\\\lambda & \mathrm{f}\mathrm{o}\mathrm{r} x=0\end{array}\right.$$and if $$\mathrm{f}$$ is continuous at $${x}=0$$ then $$\lambda=$$
  • $$-2$$
  • $$-4$$
  • $$-6$$
  • $$-8$$
lf $$f(x)=\displaystyle \frac{x(e^{1/x}-e^{-1/x})}{e^{1/x}+e^{-1/x}} x\neq 0$$ is continuous at $${x}=0$$, then $${f}(\mathrm{0})=$$
  • $$1$$
  • $$2$$
  • $$0$$
  • $$3$$
If $$f$$ : $$R\rightarrow R$$ is defined by $$f(x)=\left\{\begin{array}{ll}\dfrac{x+2}{x^{2}+3x+2} & x\in R-\{-1,-2\}\\-1 &  x=-2\\0 & x=-1\end{array}\right.$$then $$f$$ is continuous on the set:
  • $$R$$
  • $$R-\{-2\}$$
  • $$R-\{-1\}$$
  • $$R-\{-1,-2\}$$
$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { \left( 1+x \right) ^{ \dfrac { 1 }{ x }  } }{ e }  \right) ^{ \dfrac { 1 }{ sinx }  }$$ is equal to 
  • $$\sqrt { e } $$
  • e
  • $$\dfrac { 1 }{ \sqrt { e } } $$
  • 1/e
$$f(x)=\displaystyle \frac{e^{1/x^{2}}}{e^{1/x^{2}}-1}$$ , $$x\neq 0$$, $$\mathrm{f}({0})=1$$, then $$\mathrm{f}$$ at $${x}=0$$ is:
  • discontinuous
  • left continuous
  • right continuous
  • both B and C

 $$\mathrm{A}$$ function $$\mathrm{f}(\mathrm{x})$$ is defined as
$$f(x)=\left\{ \begin{matrix} ax-b & x\leq 1 \\ 3x, & 1<x<2 \\ bx^{ 2 }-a & x\geq 2 \end{matrix} \right. $$ is continuous at
$$x=1, 2$$ then:
  • $$\mathrm{a}=5,\ \mathrm{b}=2$$
  • $$\mathrm{a}=6,\ \mathrm{b}=3$$
  • $$\mathrm{a}=7,\ \mathrm{b}=4$$
  • $$\mathrm{a}=8,\ \mathrm{b}=5$$
$$f(x)=\displaystyle \min\{x,\ x^{2}\}\forall x\in R$$. Then $$f(x)$$ is
  • discontinuous at 0
  • discontinuous at 1
  • continuous on R
  • continuous at 0, 1
$$Lt_{x \rightarrow 0}\dfrac{\sin 2x+a\sin x}{x^{3}}$$ exists and finite then $$\mathrm{a}=$$
  • $$2$$
  • $$-2$$
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{-2}{3}$$
The integer $$n$$ for which $$\displaystyle \lim_{x\rightarrow 0}\frac{(\cos x-1)(\cos x-e^{x})}{x^{n}}$$ is finite non zero number is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
$$\displaystyle \lim_{x\rightarrow 1}\{1-x+[x+1]+[1-x]\}$$ , where $$[x]$$ denotes greatest integer function, is
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
Given that the function $$\mathrm{f}$$ is defined by $$f(x)=\left\{\begin{array}{l}2x-1,x>2\\k, x=2\\x^{2}-1,x<2\end{array}\right.$$is continuous at x =Then $${k}$$ is:
  • $$3$$
  • $$2$$
  • $$1$$
  • $$-3$$
lf the function $$\mathrm{f}({x})=\begin{cases}\dfrac{\sin 3x}{x} &(x\neq 0) \\ \dfrac{k}{2}&(x=0) \end{cases}$$ is continuous at $${x}=0$$, then $${k}$$ is:
  • 3
  • 6
  • 9
  • 2
The right-hand limit of the function $$\sec{x}$$ at $$\displaystyle x=-\frac { \pi  }{ 2 } $$ is
  • $$-\infty$$
  • $$-1$$
  • $$0$$
  • $$\infty$$
$$\displaystyle \lim_{x\rightarrow 0}\frac{1}{x}\cos ^{ -1 }{ \left( \frac { 1-x^{ 2 } }{ 1+x^{ 2 } }  \right)  } =$$
  • 0
  • 1
  • 2
  • does not exist
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