Limits and Continuity Test 49

Result

Limits and Continuity Test 49

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Weekly Quiz Competition

Question 1
1 / -0

Let [x] denote the greatest integer less than or equal to x. Then :
$$\underset { x\rightarrow 0 }{ lim } \dfrac { tan\left( \pi { sin }^{ 2 }x \right) +\left( \left| x \right| -sin \right) \left( x\left[ x \right] \right) ^{ 2 } }{ { x }^{ 2 } } :$$

A
does not exist

B
equal $$\pi $$

C
equal 0

D
equal $$\pi $$ + 1
Question 2
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 3
1 / -0

$$\displaystyle\lim _{ n\rightarrow \infty }{ { n }^{ 2 }\left( { x }^{ \dfrac { 1 }{ n } }-{ x }^{ \dfrac { 1 }{ n+1 } } \right) ,x>0 } $$ is equal to

A
$$0$$

B
$$e^{x}$$

C
$$log_ex$$

D
$$None\ of\ these$$

Solution
Question 4
1 / -0

If $$\displaystyle \lim _{ x\rightarrow 0 }[1+ax+bx^{2}]^{(2/x)}=e^{3}$$, then

A
$$a=3,\ b=0$$

B
$$a=\dfrac{3}{2},\ b\neq1$$

C
$$a=\dfrac{3}{2},\ b=R$$

D
$$a=2,\ b=3$$

Solution
Question 5
1 / -0

If [.] deotes the greatest integer function then
$$\begin{matrix} lim \\ x\rightarrow \pi /2 \end{matrix}\left[ \frac { x-\frac { \pi }{ 2 } }{ cosx } \right] $$ is equal to

A
$$1$$

B
$$-1$$

C
$$2$$

D
$$-2$$

Solution
Question 6
1 / -0

$$\displaystyle x\xrightarrow { lim } a\left(\sin\frac{x-a}{2}\tan\frac{\pi x}{2a} \right) $$

A
$$\displaystyle a/\pi$$

B
$$-\displaystyle a/\pi$$

C
$$\displaystyle \pi/a$$

D
$$-\displaystyle \pi/a$$

Solution
Question 7
1 / -0

$$\lim _{ x\rightarrow 0 }{ \frac { \sqrt [ 3 ]{ 1+\sin { x } } -\sqrt [ 3 ]{ 1-\sin { x } } }{ x } } =$$

A
$$0$$

B
$$1$$

C
$$\frac { 2 }{ 3 } $$

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

Solution
Question 8
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 9
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 10
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