Limits and Derivatives Test

Result

Limits and Derivatives Test - 55

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Weekly Quiz Competition

Question 1
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 2
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 3
1 / -0

Value of $$\int _{ 0 }^{ \pi /2 }{ cos3t } $$ dt is

A
$$\dfrac { 2 }{ 3 } $$

B
$$-\dfrac { 1 }{ 3 } $$

C
$$-\dfrac { 2 }{ 3 } $$

D
$$\dfrac { 1 }{ 3 } $$

Solution
Question 4
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 5
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 6
1 / -0

$$\cfrac { d }{ dx } \left[ \sin ^{ 2 }{ \left( \cos ^{ -1 }{ \sqrt { \cfrac { 1+x }{ 1-x } } } \right) } \right] =$$

A
1

B
-1

C
$$\cfrac{1}{2}$$

D
$$\cfrac{-1}{2}$$

Solution
Question 7
1 / -0

Let $$a \in \left( 0 , \frac { \pi } { 2 } \right)$$, then the value of
$$ \lim _ { a \rightarrow 0 } \frac { 1 } { a ^ { 3 } } \int _ { 0 } ^ { a } \ell n (1+tan a tan x)dx$$ is equal to

A
$$\frac { 1 }{ 3 } $$

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

C
$$\frac { 1 }{ 6 } $$

D
1
Question 8
1 / -0

$$ \lim _ { x \rightarrow 0 } \frac { \sin x } { x } = y $$

A
$$

y > 1

$$

B
$$

y < 1

$$

C
$$

y \geq 1

$$

D
$$

y \leq 1

$$

Solution
Question 9
1 / -0

Let x be an irrational, then $$\underset { m\rightarrow \infty }{ lim } \underset { n\rightarrow \infty }{ lim } \left\{ cos(n!\pi x) \right\} ^{ 2m }$$ equals

A
0

B
-1

C
1

D
Indeterminate

Solution
Question 10
1 / -0

$$\lim_{x\rightarrow 0}\frac{1-cos^{3}x}{x sib x cos x }$$ is equal to

A
2/5

B
3/5

C
3/2

D
3/4