Limits and Derivatives Test

Result

Limits and Derivatives Test - 50

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Weekly Quiz Competition

Question 1
1 / -0

$$\underset { x\rightarrow 0 }{ lim } \frac { sin({ 6x }^{ 2 }) }{ Incos({ 2x }^{ 2 }-x) } =$$

A
12

B
-12

C
6

D
-6

Solution
Question 2
1 / -0

Let $$f ( \beta ) = \lim _ { \alpha \rightarrow \beta } \dfrac { \sin ^ { 2 } \alpha - \sin ^ { 2 } \beta } { \alpha ^ { 2 } - \beta ^ { 2 } },$$ then $$f \left( \dfrac { \pi } { 4 } \right)$$ is greater than-

A
$$\lim _ { x \rightarrow 0 } \dfrac { 1 - \cos ^ { 3 } x } { x \sin 2 x }$$

B
$$\lim _ { x \rightarrow \pi / 2 } \dfrac { \cot x - \cos x } { ( \pi - 2 x ) ^ { 3 } }$$

C
$$\lim _ { x \rightarrow \infty } ( \cos \sqrt { x + 1 } - \cos \sqrt { x } )$$

D
$$\lim _ { x \rightarrow a } \dfrac { \sqrt { a + 2 x } - \sqrt { 3 x } } { \sqrt { 3 a + x } - 2 \sqrt { x } }$$ where $$a > 0$$

Solution
Question 3
1 / -0

$$\underset { x\rightarrow -\infty }{ lim } \frac { ({ 3x }^{ 4 }+{ 2x }^{ 2 })sin(\frac { 1 }{ x } )+{ |x| }^{ 3 }+5 }{ { |x }|^{ 3 }+{ |x| }^{ 2 }+|x|+1 } =$$

A
2

B
1

C
-2

D
-3
Question 4
1 / -0

The value of $$\displaystyle \lim_{x \rightarrow 0} (\sin x)^{\dfrac{1}{x}}+(1+x)^{(\sin x)})=0$$, where $$x > 0$$, is :

A
$$0$$

B
$$-1$$

C
$$1$$

D
$$2$$

Solution
Question 5
1 / -0

The value of $$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}$$ $$[\frac{x}{sinx}],$$ where [.] represents the greatest inter function , is

A
$$1$$

B
$$0$$

C
$$-1$$`

D
none of these

Solution
Question 6
1 / -0

$$\underset { x\rightarrow \frac { \pi }{ 2 } }{ lim } \frac { cotx-cosx }{ { (\pi -2x) }^{ 3 } } $$ equals :

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

B
$$\frac { 1 }{ 4 } $$

C
$$\frac { 1 }{ 24 } $$

D
$$\frac { 1 }{ 16 } $$

Solution
Question 7
1 / -0

The value of $$\displaystyle \lim_{x\rightarrow 0}\left(\dfrac {1}{x^{2}}-\cot x\right)$$ equals

A
$$1$$

B
$$0$$

C
$$\infty$$

D
$$Does\ not\ exist$$

Solution
Question 8
1 / -0

$$\underset { x\rightarrow 0 }{ Lt\quad } \frac { sec\quad x-1 }{ { \left( sec\quad x+\quad 1 \right) }^{ 2 } } =$$

A
1/8

B
11/4

C
3.2

D
2
Question 9
1 / -0

If $$u=f(x^{2}), v=g(x^{3}),f(x)=sinx, g^{1}(x)=cosx$$ then find $$\frac{du}{dv}$$

A
$$1$$

B
$$\frac{2}{3}$$

C
$$\frac{2sin x^{2}}{3xcos x^{3}}$$

D
$$\frac{2x^{2}}{3x^{3}}$$

Solution
Question 10
1 / -0

If$$y=sin x\left[ \dfrac { 1 }{ sin x sin 2 x } +\dfrac { 1 }{ sin 2 x.sin 3 x } +.....+\dfrac { 1 }{ sinn x sin\left( n+1 \right) x } \right]$$ then $$\dfrac { dy }{ dx } =$$

A
$$cot x-cot\left( n+1 \right) x$$

B
$$\left( n+1 \right) { cosec }^{ 2 }\left( n+1 \right) x-{ cosec }^{ 2 }x$$

C
$${ cosec }^{ 2 }x-\left( n+1 \right) { cosec }^{ 1 }\left( n+1 \right) x$$

D
$$cot x+cot\left( n+1 \right) x$$

Solution