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

Evaluate
limx01cos(1cos2x)x4
  • 4
  • 2
  • 1
  • 12
Evaluate the following limits.
limx2x2x2.
  • 23
  • 22
  • 25
  • None of these
If limx0x(1+acosx)bsinxx3=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
limx0  (1cos2x)22xtanxxtan2x is :
  • 2
  • 12
  • 12
  • 2
limx032x23xx is equal to
  • log32
  • 1
  • log98
  • 0
Let f(x)=ax+bx+1,limx0f(x)=2 and limxf(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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