Limits and Derivatives Test

Result

Limits and Derivatives Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

$$\underset { x\rightarrow 0 }{ Lt } \cfrac {tanx-x}{x^2tanx}$$ equals:

A
1

B
1/2

C
1/3

D
None of these

Solution
Question 2
1 / -0

$$\dfrac { d }{ dx } \left[ \left( \dfrac { { tan }^{ 2 }2x-{ tan }^{ 2 }x }{ 1-{ tan }^{ 2 }2x{ tan }^{ 2 }x } \right) cot3x \right]$$

A
$$\tan2x\tan x$$

B
$$\tan3x\tan x$$

C
$${ sec }^{ 2 }x$$

D
$$\sec x \tan x$$

Solution
Question 3
1 / -0

$$\displaystyle\lim_{x\rightarrow \infty}\left(\dfrac{x+1}{2x+1}\right)^{x^2}$$ equals?

A
$$0$$

B
e

C
$$1$$

D
$$\infty$$

Solution
Question 4
1 / -0

$$\underset { x\rightarrow \pi/2 }{ lim } \left(\dfrac{cosec x-1}{cot^2x}\right)= $$

A
$$0$$

B
$$-\dfrac{1}{2}$$

C
$$\dfrac{1}{2}$$

D
$$1$$

Solution
Question 5
1 / -0

$$\underset{x \rightarrow \infty}{lim} \dfrac{2 \tan^{-1} x}{\pi}$$ equals $$e^L$$ then $$L$$ is equal to

A
$$\dfrac{2}{\pi}$$

B
$$-\dfrac{2}{\pi}$$

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

D
$$1$$

Solution
Question 6
1 / -0

$$\displaystyle \lim _{ x\rightarrow x/2 } \dfrac { \left[ 1-\tan { \left( \dfrac { x }{ 2 } \right) } \right] \left[ 1-\sin { x } \right] }{ \left[ 1+\tan { \left( \dfrac { x }{ 2 } \right) } \right] \left[ \pi -2x \right] ^{ 3 } } $$ is

A
$$\dfrac {1}{8}$$

B
$$0$$

C
$$\dfrac {1}{32}$$

D
$$\infty$$

Solution
Question 7
1 / -0

$$\displaystyle \lim _{ \theta \rightarrow 0 }{ \frac { 4\theta \left( \tan { \theta -2\theta \tan { \theta } } \right) }{ { \left( 1-\cos { 2\theta } \right) }^{ 2 } } } $$ is

A
$$1/\sqrt {2}$$

B
$$1/2$$

C
$$1$$

D
$$2$$

Solution
Question 8
1 / -0

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { 3\sin { \left( { x }^{ 9} \right) -\sin { \left( { x }^{ 9 } \right) } } }{ { x }^{ 3 } } } =q$$

A
0

B
$$4\left(\dfrac{\pi}{200}\right)^{3}$$

C
$$\dfrac{\pi}{200}$$

D
$$\dfrac{\pi}{100}$$

Solution
Question 9
1 / -0

evaluate$$ \underset { x\rightarrow 0 }{ lim } \frac { x-\int _{ 0 }^{ x }{ { cost }^{ 2 }dt } }{ { x }^{ 3 }-6x } $$

A
$$3$$

B
$$-1$$

C
$$0$$

D
$$1$$

Solution
Question 10
1 / -0

If $$k$$ is an integer such that $$\lim_{n \rightarrow \infty} \left[\left(\cos \dfrac{k\pi}{4}\right)^{2}-\left(\cos \dfrac{k\pi}{6}\right)^{2}\right]=0$$ then :

A
$$k$$ is divisible neither by $$4$$ nor by $$8$$.

B
$$k$$ must be divisible by $$12$$ but not necessarily by $$24$$.

C
$$k$$ must be divisible by $$24$$.

D
either $$k$$ is divisible by $$24$$ or $$k$$ is neither divisible by $$4$$ nor by $$6$$.

Solution