CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 7 - MCQExams.com

Evaluate
$$\mathop {\lim }\limits_{x \to 0} \cfrac{{1 - \cos (1 - \cos 2x)}}{{{x^4}}}$$
  • $$4$$
  • $$2$$
  • $$1$$
  • $$\dfrac{1}{2}$$
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 2}\dfrac{x-2}{\sqrt{x}-\sqrt{2}}$$.
  • $$2\sqrt{3}$$
  • $$2\sqrt{2}$$
  • $$2\sqrt{5}$$
  • None of these
If $$\lim _{ x\rightarrow 0 }{ \cfrac { x\left( 1+a\cos { x }  \right) -b\sin { x }  }{ { x }^{ 3 } }  } =1$$ then
  • $$a=-5/2$$, $$b=-1/2$$
  • $$a=-3/2$$, $$b=-1/2$$
  • $$a=-3/2$$, $$b=-5/2$$
  • $$a=-5/2$$, $$b=-3/2$$
$$\displaystyle \underset { x\rightarrow 0 }{ lim } \ \ \frac { ({ 1-\cos2x) }^{ 2 } }{ 2x \tan x-x \tan2x } $$ is :
  • $$-2$$
  • $$\dfrac { -1 }{ 2 } $$
  • $$\dfrac { 1 }{ 2 } $$
  • $$2$$
$$\underset { x\rightarrow 0 }{ \lim } \dfrac { { 3 }^{ 2x }-{ 2 }^{ 3x } }{ x } $$ is equal to
  • $$\log\dfrac { 3 }{ 2 } $$
  • $$1$$
  • $$\log\cfrac { 9 }{ 8 } $$
  • $$0$$
Let $$f(x)=\dfrac{ax+b}{x+1},lim_{x\rightarrow 0} f(x)=2$$ and $$lim_{x\rightarrow \infty} f(x)=1$$ then $$f(-2)=$$
  • $$1$$
  • $$2$$
  • $$-1$$
  • $$0$$
Evaluate the limit, $$\mathop {\lim }\limits_{x \to 0} \frac{{x({{(1 + x)}^{1/x}} - e)}}{{x({{(1 + {x^2})}^{1/{x^2}}} - e)}}$$
  • 0
  • 1
  • 2
  • DNE
$$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } - x + 1 } - a x - b \right) = 0,$$   then the values of  $$a$$  and  $$b$$  are given by
  • $$a = - 1 , b = 1 / 2$$
  • $$a = 1 , b = 1 / 2$$
  • $$a = 1 , b = - 1 / 2$$
  • None of these
The value of $$\lim_{x\rightarrow \infty }$$ y In $$(\frac{sin (x+1/y)}{sin x})$$ when $$0 < x < \pi /2$$ is
  • cos x
  • $$\infty$$
  • cot x
  • does not exist
$$\lim _{ { x\rightarrow \pi /4 } } \dfrac { \cot ^{ { 3 } } x-\tan  x }{ \cos  (x+\pi /4) } $$  is
  • $$4$$
  • $$8 \sqrt { 2 }$$
  • $$8$$
  • $$4 \sqrt { 2 }$$
$$\displaystyle\lim_{n\rightarrow\infty}\left\{\dfrac{n!}{(kn)^n}\right\}^{\dfrac{1}{n}}, k\neq 0$$, is equal to?
  • $$\dfrac{k}{e}$$
  • $$\dfrac{e}{k}$$
  • $$\dfrac{1}{ke}$$
  • None of these
Let $$f(x)=\displaystyle\lim_{n\rightarrow \infty}\sum^{n-1}_{r=0}\dfrac{x}{(rx+1)\{(r+1)x+1\}}$$, then?
  • $$f(x)$$ is continuous but not differentiable at $$x=0$$
  • $$f(x)$$ is both continuous and differentiable at $$x=0$$
  • $$f(x)$$ is neither continuous nor differentiable at $$x=0$$
  • $$f(x)$$ is a periodic function
$$\underset { x\rightarrow 0 }{ lim } \frac { tan(sinx)-x }{ { tanx }^{ 3 } } $$ is equal to 
  • $$\frac { 1 }{ 6 } $$
  • $$\frac { 1 }{ 3 } $$
  • $$\frac { 1 }{ 2 } $$
  • 1
$$\displaystyle \lim _{ x\rightarrow \infty }{ \left[\dfrac{n}{n^{2}+1^{2}}+\dfrac{n}{n^{2}+2^{2}}+\dfrac{n}{n^{2}+3^{2}}+....+\dfrac{1}{n^{5}}\right] }$$
  • $$\pi/4$$
  • $$\tan^{-1}{(2)}$$
  • $$\pi/2$$
  • $$\tan^{-1}{(3)}$$
The value of $$\displaystyle n\xrightarrow { lim } \infty\frac{1.n+2.(n-1)+3.(n-2)+...+n.1}{{1}^{2}+{2}^{2}+...+{n}^{2}}$$ is
  • $$1$$
  • $$-1$$
  • $$\displaystyle \frac{1}{\sqrt{2}}$$
  • $$\displaystyle \frac{1}{2}$$
$$\displaystyle\lim _{ x\rightarrow 0 }{ { x }^{ 2 }{ e }^{ \sin { \frac { 1 }{ x }  }  } } $$ equals 
  • $$1$$
  • $$0$$
  • $$\infty$$
  • Does not exist
Evaluate: $$\underset { { x\rightarrow}\dfrac{ \pi  }{ 4 }  }{ lim } \dfrac { { cot }^{ 3 }x-tanx }{ cos\left( x+\pi /4 \right)  } \quad $$
  • $$8$$
  • $$8\sqrt { 2 } $$
  • $$4$$
  • $$4\sqrt { 2 } $$
$${ lim }_{ x\rightarrow 1 }(1+cos\pi ){ cot }^{ 2 }\pi x=-----$$
  • 1
  • -1
  • $$\dfrac { 1 }{ 2 } $$
  • 0
Let $$f:(0, \infty)\to R$$ be a differentiable function such that $$f'(x)=2-\dfrac{f(x)}{x}$$ for all $$x\in (0, \infty)$$ and $$f(1)\neq 1$$. Then 
  • $$\underset { x\rightarrow { 0 }^{ + } }{ \lim } f'\left( \dfrac { 1 }{ x } \right) =1$$
  • $$\underset { x\rightarrow { 0 }^{ + } }{ \lim } xf\left( \dfrac { 1 }{ x } \right) =2$$
  • $$\underset { x\rightarrow { 0 }^{ + } }{ \lim } x^{ 2 }f'\left( x \right) =0$$
  • $$\left| f\left( x \right) \right| \le 2$$ for $$ $$ all $$X\in \left( 0,2 \right) $$
$$\displaystyle\lim_{x\rightarrow \dfrac{\pi}{2}}\dfrac{\cot x-\cos x}{(\pi -2x)^3}$$ equals?
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{24}$$
  • $$\dfrac{1}{16}$$
  • $$\dfrac{1}{8}$$
Let $$f : R \to R$$ be a differentiable function satisfying $$f'(3) + f'(2) = 0$$.
Then $$\underset{x \to 0}{\lim} \left(\dfrac{1+f(3+x)-f(3)}{1+f(2-x) - f(2)}\right)^{\frac{1}{x}}$$ is equal to 
  • $$e^2$$
  • $$e$$
  • $$e^{-1}$$
  • $$1$$
If the function $$f(x)$$ satisfies the relation $$f(x+y)=y\dfrac{|x-1|}{(x-1)}f(x)+f(y)$$ with $$f(1)=2$$, then $$\displaystyle\lim_{x\rightarrow 1}f'(x)$$ is?
  • $$2$$
  • $$-2$$
  • $$0$$
  • Limit do not exixst
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow a}\dfrac{\sqrt{x}+\sqrt{a}}{x+a}$$.
  • $$-\dfrac{1}{\sqrt{a}}$$
  • $$\dfrac{1}{{a}}$$
  • $$\dfrac{1}{2\sqrt{a}}$$
  • $$\dfrac{1}{\sqrt{a}}$$
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{x^{2/3}-9}{x-27}$$.
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{5}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{3x+1}{x+3}$$.
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{5}{3}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 1}\dfrac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}}, x > 1$$.
  • $$\dfrac{\sqrt{2}+1}{\sqrt{2}}$$
  • $$\dfrac{\sqrt{2}-1}{\sqrt{2}}$$
  • $$\dfrac{\sqrt{2}+1}{{2}}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{ax+b}{cx+d}, d\neq 0$$.
  • $$\dfrac{a}{c}$$
  • $$\dfrac{a}{d}$$
  • $$\dfrac{b}{d}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{a^2+x^2}-a}{x^2}$$.
  • $$\dfrac{1}{\sqrt a}$$
  • $$\dfrac{1}{\sqrt {2a}}$$
  • $$\dfrac{1}{a}$$
  • $$\dfrac{1}{2a}$$
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{2x}{\sqrt{a+x}-\sqrt{a-x}}$$.
  • $$-2\sqrt{a}$$
  • $$\sqrt{a}$$
  • $$2\sqrt{a}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a^2+ax}}$$.
  • $$\dfrac{1}{2\sqrt{a}}$$
  • $$\dfrac{1}{2a\sqrt{a}}$$
  • $$\dfrac{1}{2a}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 4}\dfrac{2-\sqrt{x}}{4-x}$$.
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{3}$$
  • None of these
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 2}\dfrac{\sqrt{1+4x}-\sqrt{5+2x}}{x-2}$$.
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{5}$$
Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{2-x}-\sqrt{2+x}}{x}$$.
  • $$\dfrac{1}{\sqrt{2}}$$
  • $$-\dfrac{1}{\sqrt{3}}$$
  • $$-\dfrac{1}{{2}}$$
  • $$-\dfrac{1}{\sqrt{2}}$$
Evaluate the following limits.
If $$\displaystyle\lim_{x\rightarrow a}\dfrac{x^5-a^5}{x-a}=405$$, find all possible values of a.
  • $$a=3, -3$$
  • $$a=2, -2$$
  • $$a=5, -5$$
  • None of these
Evaluate the following limits.
If $$\displaystyle\lim_{x\rightarrow a}\dfrac{x^9-a^9}{x-a}=9$$, find all possible values of a.
  • $$2, -2$$.
  • $$1, -1$$.
  • $$1, 0$$.
  • None of these
Evaluate the following limit.
$$\displaystyle\lim_{x\rightarrow 0}\dfrac{8^x-2^x}{x}$$.
  • $$log 4$$
  • $$log 6$$
  • $$log 5$$
  • None of these
If $$f : R \rightarrow (0, \infty)$$ is an increasing function and if $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(3x)}{f(x)} = 1$$, then $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(2x)}{f(x)}$$ is equal to 
  • $$\dfrac{2}{3}$$
  • $$\dfrac{3}{2}$$
  • $$2$$
  • $$3$$
  • $$1$$
$$\underset{x\to 0}{\lim} \left(\dfrac{3x^2+2}{7x^2+2}\right)^{1/x^2}$$ is equal to:
  • $$\dfrac{1}{e^2}$$
  • $$\dfrac{1}{e}$$
  • $$e^2$$
  • $$e$$
If $$f$$ is differentiable at $$x = 1$$ and $$\underset{h \rightarrow 0}{\lim} \dfrac{1}{h} f (1 + h) = 5, f'(1) = $$
  • $$0$$
  • $$1$$
  • $$3$$
  • $$4$$
  • $$5$$
The value of $$ \displaystyle \lim _{x \rightarrow \pi} \dfrac{1+\cos ^{3} x}{\sin ^{2} x} $$ is
  • 1/3
  • 2/3
  • -1/4
  • 3/2
$$\displaystyle \lim _{x \rightarrow 0} \dfrac{x\left(e^{x}-1\right)}{1-\cos x} $$ is equal to
  • 0
  • $$\infty$$
  • -2
  • 2
$$ \displaystyle \lim _{x \to \pi / 2}\left[x \tan x-\left(\dfrac{\pi}{2}\right) \sec x\right] $$ is equal to 
  • 1
  • -1
  • 0
  • $$None \ of \ these$$
$$ \displaystyle \lim _{x \rightarrow-\infty} \dfrac{x^{2} \tan \dfrac{1}{x}}{\sqrt{8 x^{2}+7 x+1}} $$ is equal to
  • $$ -\dfrac{1}{2 \sqrt{2}} $$
  • $$ \dfrac{1}{2 \sqrt{2}} $$
  • $$ \dfrac{1}{\sqrt{2}} $$
  • Does not exist
$$\displaystyle\lim_{n\rightarrow \infty}\dfrac{(n 1)^{1/n}}{n}$$ equals?
  • $$1$$
  • $$e$$
  • $$e^{-1}$$
  • None of these
If $$ f(x)=\dfrac{\cos x}{(1-\sin x)^{1 / 3}}, $$ then
  • $$\lim _{ x\rightarrow \dfrac { \pi^- }{2 } }{ f(x)=-\infty } $$
  • $$\lim _{ x\rightarrow \dfrac { \pi^+ }{2 } }{ f(x)=\infty } $$
  • $$\lim _{ x\rightarrow \dfrac { \pi }{2 } }{ f(x)=\infty } $$
  • none of these
$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x^{n}}{(\sin x)^{m}},(m<n) $$ is equal to
  • 1
  • 0
  • n/m
  • None of these
$$\displaystyle  \lim _{x \rightarrow 1} \dfrac{1+\sin \pi\left(\dfrac{3 x}{1+x^{2}}\right)}{1+\cos \pi x} $$ is equal to
  • 0
  • 1
  • 2
  • 4
The value of $$ \displaystyle \lim _{x \rightarrow 2} \dfrac{2^{x}+2^{3-x}-6}{\sqrt{2^{-x}}-2^{1-x}} $$ is
  • 16
  • 8
  • 4
  • 2
 $$ The \ value \ of  \displaystyle \lim _{x \rightarrow 1}(2-x)^{\tan \dfrac{\pi x}{2}} $$ is
  • $$e^{-2 \pi} $$
  • $$ e^{1 / \pi} $$
  • $$ e^{2 /\pi} $$
  • $$ e^{-1 / \pi} $$
$$\displaystyle \lim _{x \rightarrow 1} \dfrac{1-x^{2}}{\sin 2 \pi x} \text { is equal to }$$
  • $$ \dfrac{1}{2 \pi} $$
  • $$ \dfrac{-1}{\pi} $$
  • $$ \dfrac{-2}{\pi} $$
  • None of these
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