CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 5 - MCQExams.com

The limit of $$x\sin { \left( { e }^{ \frac { 1 }{ x }  } \right)  } $$ as $$x\rightarrow 0$$
  • is equal to $$0$$
  • is equal to $$1$$
  • is equal to $$\cfrac { e }{ 2 } $$
  • does not exist
The function represented by  the following graph is.

572668_db4e79fb2abd478fa7332735dca6ad0f.png
  • Differentiable but not continuous $$x=1$$
  • Neither continuous nor Differentiable at $$x=1$$
  • Continuous but not Differentiable at $$x=1$$
  • Continuous but Differentiable at $$x=1$$
If [x] denotes the greatest integer not exceeding $$x$$ and if the function $$f$$ defined by $$f(x)= \begin{cases}\dfrac{a+2\cos\,x}{x^2}&(x < 0) \\ b\,\tan \dfrac{\pi}{[x+4]}&(x \ge 0) \end{cases}$$ is continuous at $$x=0$$, then the order pair (a, b) =
  • $$(-2, 1)$$
  • $$(-2, -1)$$
  • $$(-1, \sqrt{3})$$
  • $$(-2,-\sqrt{3})$$
$$\displaystyle \lim_{x\rightarrow 0}\dfrac {1 - \cos x}{x^{2}}$$ is ____
  • $$2$$
  • $$3$$
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{3}$$
If $$\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^3+1}{x^2+1}-(ax+b)=2$$, then
  • $$a=2$$ and $$b=-1$$
  • $$a = 1$$ and $$b = 1$$
  • $$a = 1$$ and $$b = -1$$
  • $$a = 1$$ and $$b = -2$$
$$\lim _ { x \rightarrow 0 } \dfrac { ( 27 + x ) ^ { 1 / 3 } - 3 } { 9 - ( 27 + x ) ^ { 2 / 3 } }$$  equals :
  • $$- 1 / 6$$
  • $$1 / 6$$
  • $$1 / 3$$
  • $$- 1 / 3$$
The value of the constant $$\alpha$$ and $$\beta$$ such that $$\displaystyle \lim_{x\rightarrow \infty}\left(\displaystyle\frac{x^2+1}{x+1}-\alpha x-\beta\right)=0$$ are respectively.
  • $$(1, 1)$$
  • $$(-1, 1)$$
  • $$(1,-1)$$
  • $$(0, 1)$$
What is $$\displaystyle \lim_{x \rightarrow 0 }  x^2 \sin \left(\frac{1}{x}\right)$$ equal to ? 
  • 0
  • 1
  • 1/2
  • Limit does not exist.
If the function $$f(x)$$ satisfies $$\displaystyle \lim_{x\rightarrow 1}\frac{f(x)-2}{x^2-1}=\pi$$, then $$\displaystyle \lim_{x\rightarrow 1}f(x)=$$
  • $$2$$
  • $$3$$
  • $$1$$
  • $$0$$
The limit of $$\left[\frac{1}{x^2}+\frac{(2013)^x}{e^x-1}-\frac{1}{e^x-1}\right]$$ as $$x\rightarrow 0$$
  • Approaches $$+\, \infty$$
  • Approaches $$- \,\infty$$
  • Is equal to $$log_e(2013)$$
  • Does not exist
$$\displaystyle \lim_{x\rightarrow 0}\frac{log_e(1+x)}{3^x-1}=$$ ____.
  • $$log_e3$$
  • $$0$$
  • $$log_3 e$$
  • $$1$$
The limit of $$\displaystyle \sum_{n=1}^{1000}(-1)^nx^n$$ as $$x\rightarrow \infty$$
  • does not exist
  • exists and equals to 0
  • exists and approaches $$+\infty$$
  • exists and approaches $$-\infty$$
Which one of the following statements is correct?
  • $$\displaystyle \lim_{x \rightarrow 0} (fog) (x)$$ exists.
  • $$\displaystyle \lim_{x \rightarrow 0} (gof) (x)$$ exists.
  • $$\displaystyle \lim_{x \rightarrow 0-} (fog) (x) = \displaystyle \lim_{x \rightarrow 0-} (gof) (x)$$
  • $$\displaystyle \lim_{x \rightarrow 0+} (fog) (x) =\displaystyle \lim_{x \rightarrow 0-} (gof) (x)$$
The limit of $$\left\{\frac{1}{x}\sqrt{1-x}-\sqrt{1+\frac{1}{x^2}}\right\}$$ as $$x\rightarrow 0$$
  • Does not exist
  • Is equal to $$\frac{1}{2}$$
  • Is equal to 0
  • Is equal to 1
$$\displaystyle\lim_{x\rightarrow\frac{\pi}{6}}\frac{\sin\left(x-\displaystyle\frac{\pi}{6}\right)}{\sqrt{3-2cos x}}$$ is equal to :
  • $$0$$
  • $$\displaystyle\frac{1}{(\sqrt{3}-2)}$$
  • $$1$$
  • $$\infty$$
Evaluate: $$\displaystyle\lim_{x\to 10}\dfrac{x^2-100}{x-10}$$
  • $$10$$
  • $$-5 $$
  •  $$20$$
  • $$5$$
$$ \underset { x\rightarrow 0 }{ lim } \cfrac { { \left( 25 \right)  }^{ x }-2\left( 15 \right)^ x+{ 9 }^{ x } }{ cos6x-cos2x }  $$ is equal to :
  • $$ log \left ( \cfrac {5} {3} \right ) $$
  • $$ \cfrac {1} {4} log 15 $$
  • $$ - \cfrac {1} {16} \left( \cfrac {5} {3} \right) ^2 $$
  • $$log \left ( \cfrac {3} {5} \right ) $$
$$\displaystyle \lim_{x\rightarrow \pi/4} \dfrac {\tan x - 1}{\cos 2x}$$ is equal to
  • $$1$$
  • $$0$$
  • $$-2$$
  • $$-1$$
If $$\underset{x\to 0}{\lim}\dfrac{x^a\sin^b x}{\sin(x^c)}, a, b, c, \in R \sim \{0\}$$ exists and has non-zero value, then 
  • $$a,b,c$$ are in A.P
  • $$a,b,c$$ are in G.P
  • $$a,b,c$$ are in H.P
  • none of these
$$\displaystyle \lim_{x\rightarrow 3} = \dfrac {\sqrt {x} -\sqrt {3}}{\sqrt {x^{2} - 9}}$$ is equal to
  • $$1$$
  • $$3$$
  • $$\sqrt {3}$$
  • $$-\sqrt {3}$$
  • $$0$$
The value of $$\displaystyle \lim _{ x\rightarrow \pi /6 }{ \cfrac { 2\sin ^{ 2 }{ x } +\sin { x } -1 }{ 2\sin ^{ 2 }{ x } -3\sin { x } -1 }  } $$ is
  • $$3$$
  • $$-3$$
  • $$6$$
  • $$0$$
If $$f\left( x \right) =\begin{vmatrix} \sin { x }  & \cos { x }  & \tan { x }  \\ { x }^{ 3 } & { x }^{ 2 } & x \\ 2x & 1 & 1 \end{vmatrix}$$, then $$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { f\left( x \right)  }{ { x }^{ 2 } }  } $$ is
  • $$-1$$
  • $$3$$
  • $$1$$
  • Zero
If $$\displaystyle \lim _{ n\rightarrow \infty  }{ \cfrac { n.{ 3 }^{ n } }{ n{ \left( x-2 \right)  }^{ n }+n.{ 3 }^{ n+1 }-{ 3 }^{ n } }  } =\cfrac { 1 }{ 3 } $$, then the range of $$x$$ is (When $$n\in N$$)
  • $$[2,\ 5)$$
  • $$\left( 1,\ 5 \right) $$
  • $$\left( -1,\ 5 \right) $$
  • $$\left( -\infty ,\ \infty \right) $$
$$\lim _{ x\rightarrow 3 }{ \left( { x }^{ 3 }-4 \right) /\left( x+1 \right)  } =$$
  • $$(4/23)$$
  • $$(2/23)$$
  • $$(1/8)$$
  • $$(23/4)$$
$$\displaystyle\lim_{x\rightarrow \frac{\pi}{2}}(\pi - 2x^{\cos x})$$ is equal to :
  • $$\pi+2$$
  • $$\pi-2$$
  • e
  • $$e^{-1}$$
Determine the value of k for which the following function is continuous at $$x=3$$.
$$f(x)=\dfrac{x^2-9}{x-3}, x \neq 3$$

$$f(x)=k, x=3$$
  • 2
  • 4
  • 6
  • 8
If $$f(x) = \begin{vmatrix} \cos x& x & 1\\ 2\sin x & x^{2} & 2x\ \\ \tan x & x & 1\end{vmatrix}$$, then $$\displaystyle \lim_{x\rightarrow 0} \dfrac {f'(x)}{x}$$.
  • Exists and is equal to $$-2$$
  • Does not exist
  • Exist and is equal to $$0$$
  • Exists and is equal to $$2$$
Which of the following statements are true for the function $$f(x)$$ defined for $$-1\leq x \leq 3$$ in the figure shown?

880039_ce1751dc863440c2a77ba31deb50b922.png
  • $$\lim_\limits{x\to-1^+} f(x) = 1$$
  • $$\lim_\limits{x\to2} f(x) $$ does not exist
  • $$\lim_\limits{x\to1^-} f(x) = 2$$
  • $$\lim_\limits{x\to 0^+} f(x) =\lim_\limits{x\to 0^-} f(x) $$
$$\dfrac{\displaystyle \lim_{h \rightarrow 0}(h+1)^2}{\displaystyle \lim_{h\rightarrow 0}(1+h)^{2/h}}$$ is equal to
  • $$e^1$$
  • $$e^{-2}$$
  • $$e^2$$
  • $$e^{-1}$$
$$\mathop {\lim }\limits_{x \to 0} {{(1 - \cos 2x)(3 + \cos x)} \over {x\tan 4x}}$$ is equal to

  • 2
  • 1/2
  • 4
  • 3
Evaluate the limit:
$$\displaystyle \lim_{x \to 0} \left( \frac{x- \sin x}{x}\right) \sin \left(\frac{1}{x} \right)$$
  • $$1$$
  • $$-1$$
  • Does not exist
  • $$0$$
$$\lim _{ x\rightarrow { 0 }^{ + } }{ \left( { \left( x\cos { x }  \right)  }^{ x }+{ \left( \cos { x }  \right)  }^{ \frac { 1 }{ \ln { x }  }  }+{ \left( x\sin { x }  \right)  }^{ x } \right)  } $$ is equal to
  • $$2$$
  • $$2+e$$
  • $$2+\dfrac { 1 }{ e }$$
  • $$3$$
If the function $$f(x)=\dfrac{e^{x^{2}}-\cos x}{x^{2}}$$ for $$x \neq 0$$ continuous at $$x=0$$ then $$f(0)=$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{2}$$
  • $$2$$
  • $$\dfrac{1}{3}$$

The function $$f : R /{0} \rightarrow R$$ given by $$f(x) =
\dfrac{1}{x} - \dfrac{2}{e^{2x} -1}$$ can be made continuous at $$x=0$$ by
defining $$f(0)$$ as 
  • 0
  • 1
  • 2
  • -1
$$\lim\limits_{x\to 0}\dfrac{e^{\sin x}-1}{x}=$$
  • $$0$$
  • $$1$$
  • $$-1$$
  • none of these
Let $${P_n} = \prod\limits_{k = 2}^n {\left( {1 - {1 \over {{}^{{}^{k + 1}}{C_2}}}} \right)} .$$ If $$\mathop {\lim }\limits_{x \to \infty } {P_n}$$ can be expressed as lowest rational in the form $$\dfrac { a }{ b } $$ , then value of $$(a+b)$$ is __________.
  • 12
  • 4
  • 8
  • 10
If $$\mathop {\lim }\limits_{x \to 0} {\left( {\cos x + a\sin bx} \right)^{\frac{1}{x}}} = {e^2}$$ then
the possible values of $$'a'\& 'b'are:$$ 
  • $$a = 1,b = 2$$
  • $$a = 2,b = 1$$
  • $$a = 3,b = 2/3$$
  • $$a = 2/3,b = 3$$
$$\mathop {\lim}\limits_{x \to \frac{\pi}{2}} \tan x = $$
  • $$\infty $$
  • $$ - \infty $$
  • $$0$$
  • does not exist
Suppose the function $$f(x)-f(2x)$$ has the derivative $$5$$ at $$x=1$$ and derivative $$7$$ at $$x=2$$. The derivative of the function $$f(x)-f(4x)$$ at $$x=1$$, has the value equal to?
  • $$19$$
  • $$9$$
  • $$17$$
  • $$14$$
$$\lim _{ x\rightarrow 0 }{ \log _{ \left( \tan ^{ 2 }{ x }  \right)  }{ \left( \tan ^{ 2 }{ 2x }  \right) = }  }$$
  • $$1$$
  • $$2$$
  • $$\dfrac {1}{2}$$
  • $$Does\ not\ exist$$
The value of $$\displaystyle \lim_{x\rightarrow 0}\dfrac {\sqrt {1-\cos 2x}}{x}$$ equals
  • $$0$$
  • $$1$$
  • $$\sqrt {2}$$
  • $$Does\ not\ exist$$
If $$ l = \lim_{x
\rightarrow 0} \dfrac{ x(1+ a\ cos x) - b\ sinx}{x^3} $$ is finite, 
 where $$ l \in R$$, then 
  • (a)$$(a,b) = (-1,0)$$
  • (b) a & b are any real numbers
  • (c)$$l = 0$$
  • (d)$$ l = \frac{1}{2}$$
$$\underset{h \rightarrow 0}{lim} \dfrac{\sqrt{x + h} -\sqrt{x}}{h}$$ is equal to 
  • $$\sqrt{x}$$
  • $$\dfrac{1}{2 \sqrt{x}}$$
  • $$2 \sqrt{x}$$
  • $$\dfrac{1}{\sqrt{x}}$$
Evaluate:
$$\lim\limits_{x\to 0}(1+ax)^{\dfrac{1}{x}}$$
  • $$e^a$$
  • $$e^{\dfrac{1}{a}}$$
  • $$1$$
  • None of these
The value of $$\lim\limits_{x\to 0}\dfrac{1-\cos x}{x^2}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{4}$$
  • $$2$$
  • none of these
$$\displaystyle\mathop {\lim }\limits_{\ x \to 0} \cos \frac{x}{2}\cos \frac{x}{{{2^2}}}\cos \frac{x}{{{2^3}}}......\cos \frac{x}{{{2^n}}}$$ is equal to 
  • $$1$$
  • $$-1$$
  • $$\dfrac{{\sin x}}{x}$$
  • $$\dfrac{x}{{\sin \,x}}$$
$$\lim\limits_{x\to 0}\dfrac{1-\cos x }{x^2}=$$
  • $$4$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • $$1$$
$$ \lim _{ x\rightarrow 1 }{ \dfrac { \sqrt { 1-\cos { 2\left( x-1 \right)  }  }  }{ x-1 }  }$$
  • Exists and it equals $$\sqrt {2}$$
  • Exists and it equals $$-\sqrt {2}$$
  • Does not exist because $$x-1\rightarrow 0$$
  • Does not exist because left hand limit is not equal to right hand limit
$$\underset{x \to 0}{\lim}\dfrac{\sin [ \cos x]}{1+[\cos x]}$$ is
  • $$1$$
  • $$0$$
  • does not exist
  • $$2$$
$$\mathop {\lim }\limits_{x \to 0} \dfrac{{{e^{4x}} - 1}}{x}$$
  • $$1$$
  • $$3$$
  • $$4$$
  • $$2$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers