Limits and Continuity Test 55

Result

Limits and Continuity Test 55

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Weekly Quiz Competition

Question 1
1 / -0

$$\lim _{ x\rightarrow 0 }{ \frac { 1 }{ { x }^{ 8 } } [1-\cos { (\frac { { x }^{ 2 } }{ 2 } ) } ] } [1-\cos { (\frac { { x }^{ 2 } }{ 4 } ) } ]$$

A
$$\frac { 1 }{ 8 } $$

B
$$\frac { 1 }{ { 8 }^{ 2 } } $$

C
$$\frac { 1 }{ { 8 }^{ 3 } } $$

D
$$\frac { 1 }{ { 8 }^{ 4 } } $$

Solution
Question 2
1 / -0

$$\underset { x\rightarrow a }{ lim } \left( \sin { \dfrac { x-a }{ 2 } \tan { \dfrac { \pi x }{ 2a } } } \right)$$

A
$$a/\pi$$

B
$$-a/\pi$$

C
$$\pi/a$$

D
$$-\pi/a$$

Solution
Question 3
1 / -0

The value of $$\mathop {{\text{Limit}}}\limits_{x \to 0} \frac{{\cos \left( {\sin x} \right) - \cos x}}{{{x^4}}}$$ is equal to

A
1/5

B
1/6

C
1/4

D
1/2

Solution
Question 4
1 / -0

$$\displaystyle \lim _{ x\rightarrow 0 }{ \left[ \dfrac { 100\tan { x } .\sin { x } }{ { x }^{ 2 } } \right] } $$ where $$[.]$$ represents greatest integer function is

A
$$99$$

B
$$100$$

C
$$0$$

D
$$98$$

Solution
Question 5
1 / -0

The value of $$ \lim _ { x \rightarrow 0 } \dfrac { \sin^3 ( \sqrt { x } ) \ln ( 1 + 3 x ) } { \left( \tan ^ { - 1 } \sqrt { x } \right) ^ { 2 } \left( e ^ { 5 ( \sqrt { x } ) } - 1 \right)x } $$ is equal to

A
$$

\frac { 1 } { 5 }

$$

B
$$

\frac { 3 } { 5 }

$$

C
$$

\frac { 2 } { 5 }

$$

D
$$

\frac { 4 } { 5 }

$$
Question 6
1 / -0

The value of $$\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \tan ^{ 2 }{ \left( \sqrt { 2\sin ^{ 2 }{ x } +3\sin { x } +4 } -\sqrt { \sin ^{ 2 }{ x } +6\sin { x } +2 } \right) } } $$ is equal to

A
0

B
$$\cfrac{1}{11}$$

C
$$\cfrac{1}{12}$$

D
$$\cfrac{1}{8}$$

Solution
Question 7
1 / -0

$$ \lim _{x \rightarrow 0}\left(\frac{e^{x}+e^{-x}-2}{x^{2}}\right)^{1 / x^{2}} $$

A
$$

e^{1 / 2}

$$

B
$$

e^{1 / 4}

$$

C
$$

e^{1 / 3}

$$

D
$$

e^{1 / 12}

$$

Solution
Question 8
1 / -0

$$\mathop {Lt}\limits_{x \to 1} {\left( {1 - x} \right)^{\tan \pi x}} = $$

A
$$-1$$

B
$$0$$

C
$$1$$

D
$$2$$

Solution
Question 9
1 / -0

$$\begin{matrix} lim \\ x\rightarrow a \end{matrix}(2-\frac { a }{ x } ){ }^{ tan(\frac { \pi x }{ 2a } ) }$$ is equal to

A
$$ e^{-\frac{a}{\pi}}$$

B
$$ e^{-\frac{2a}{\pi}}$$

C
$$ e^{-\frac{2}{\pi}}$$

D
1
Question 10
1 / -0

The value of $$\displaystyle \lim _{ x\rightarrow 0 }\log_{\cos{2x}}{\cos{x}}+\log_{\cos{2x}}{\cos{2x}}$$ equals

A
$$\dfrac{5}{4}$$

B
$$\dfrac{17}{4}$$

C
$$\dfrac{13}{16}$$

D
$$\dfrac{29}{10}$$

Solution