Limits and Derivatives Test

Result

Limits and Derivatives Test - 56

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Weekly Quiz Competition

Question 1
1 / -0

$$\underset { \rightarrow }{ Lim } \frac { 1-{ cos }^{ 2 }x }{ xsin2x } $$

A
1/2

B
3/2

C
3/4

D
1/4
Question 2
1 / -0

$$\displaystyle x\xrightarrow { lim } 5\quad \left(\frac{\sqrt{1-\cos(2x-10)}}{\sin (x-5)} \right) $$

A
$$-\sqrt{2}$$

B
$$\sqrt{2}$$

C
does not exist

D
none of these

Solution
Question 3
1 / -0

$$\lim _ { x \rightarrow 0 } \int _ { 0 } ^ { x } \dfrac { \left( \tan ^ { - 1 } t \right) ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } d t$$ is equal to

A
$$\pi ^ { 2 }$$

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

C
$$\dfrac { \pi ^ { 2 } } { 4 }$$

D
None of these

Solution
Question 4
1 / -0

$$\underset {x\rightarrow 0}{Lim} \frac {sin x}{x} = y $$

A
$$y>1$$

B
$$y<1$$

C
$$y\ge 1$$

D
$$y \le 1$$

Solution
Question 5
1 / -0

If $$y=e^{\sin^{2}x +\sin^{4}x + \sin ^{6}x +....+\infty} $$ , then $$\dfrac {dy}{dx} =?$$

A
$$2~\tan x~\sec^2x~e^{\tan^2~x}$$

B
$$2~\sec^2x~e^{\tan^2~x}$$

C
$$\sec^2x~e^{\tan^2~x}$$

D
$$\tan x~\sec x~e^{\tan^2~x}$$
Question 6
1 / -0

The value of $$\lim _{ x\rightarrow 0 }{ \left( { \left( \sin { x } \right) }^{ 1/x }+{ \left( \dfrac { 1 }{ x } \right) }^{ \sin { x } } \right) } $$

A
$$0$$

B
$$1$$

C
$$\infty$$

D
$$-1$$

Solution
Question 7
1 / -0

Find:
$$\underset { x\rightarrow 0 }{ lim } \quad \dfrac { 1-cos^{ 3 }x }{ xsin2x } =\quad $$

A
1/2

B
3/2

C
3/4

D
1/4
Question 8
1 / -0

$$\displaystyle \lim_{x\rightarrow 0}\dfrac {\sin x - x}{x^{3}}$$ is equal to

A
$$-\dfrac {1}{6}$$

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

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

D
$$-1$$

Solution
Question 9
1 / -0

$$\lim _{ x\rightarrow 0 }{ \dfrac { 1-\cos { x } }{ { { x\log { (1+x) } } } } } $$ =

A
1

B
0

C
-1

D
1/2

Solution
Question 10
1 / -0

The value of $$\mathop {\lim }\limits_{x \to 0} \frac{{\sec 5x - \sec 3x}}{{\sec 3x - \sec x}}$$

A
-2

B
1

C
2

D
-1/2

Solution