Limits and Continuity Test 39

Result

Limits and Continuity Test 39

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Weekly Quiz Competition

Question 1
1 / -0

$$\lim_{n\rightarrow \infty}\dfrac{1}{n^{2}}\left[\sin^{3}\dfrac{\pi}{4n}+2\sin^{3}\dfrac{2\pi}{4n}+3\sin^{3}\dfrac{3\pi}{4n}+....+n\sin^{3}\dfrac{n\pi}{4n}\right]=$$

A
$$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52-15\pi\right)$$

B
$$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+15\pi\right)$$

C
$$\dfrac{\sqrt{2}}{9\pi}\left(52-17\pi\right)$$

D
$$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+17\pi\right)$$

Solution
Question 2
1 / -0

If $$ \alpha \quad and \beta $$ are the roots of the equation $$ {ax}^{2}+bx+c=0 $$, then
$$ \underset { x\rightarrow \cfrac { \pi }{ 2 } }{ lim } \cfrac { tan\left[ \left( \alpha +\beta \right) x \right] }{ sin\left[ \left( \alpha \beta \right) x \right] } $$ is equal to :

A
$$ \cfrac {c} {b} $$

B
$$ - \cfrac {b} {c} $$

C
$$ \cfrac {a} {b} $$

D
$$- \cfrac {a} {b} $$

Solution
Question 3
1 / -0

$$\begin{matrix} lim \\ n\rightarrow \infty \end{matrix}\int _{ 0 }^{ 1 }{ \frac { { nx }^{ n-1 } }{ 1+{ x }^{ 2 } } } dx=$$

A
0

B
1

C
2

D
$$\frac { 1 }{ 2 } $$

Solution
Question 4
1 / -0

If $$L = \underset{x \rightarrow 0}{lim} \dfrac{a \, sin \, x - sin \, 2x}{tan^3 x}$$ is finite, then the value of L is :

A
1

B
2

C
3

D
-1

Solution
Question 5
1 / -0

$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { \left( 1+x \right) ^{ \dfrac { 1 }{ x } } }{ e } \right) ^{ \dfrac { 1 }{ sinx } }$$ is equal to

A
$$\sqrt { e } $$

B
e

C
$$\dfrac { 1 }{ \sqrt { e } } $$

D
1/e
Question 6
1 / -0

$$ \underset { x\rightarrow 0 }{ lim } \cfrac { { \left( 25 \right) }^{ x }-2\left( 15 \right)^ x+{ 9 }^{ x } }{ cos6x-cos2x } $$ is equal to :

A
$$ log \left ( \cfrac {5} {3} \right ) $$

B
$$ \cfrac {1} {4} log 15 $$

C
$$ - \cfrac {1} {16} \left( \cfrac {5} {3} \right) ^2 $$

D
$$log \left ( \cfrac {3} {5} \right ) $$
Question 7
1 / -0

$$Lt_{x\to 0} \dfrac{sin x - x+\dfrac{x^3}{6}}{x^5}=$$_________

A
$$\dfrac{1}{120}$$

B
$$\dfrac{-1}{120}$$

C
$$0$$

D
$$\dfrac{1}{6}$$

Solution
Question 8
1 / -0

$${ lim }_{ x\rightarrow 1 }(1+cos\pi ){ cot }^{ 2 }\pi x=-----$$

A
1

B
-1

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

D
0

Solution
Question 9
1 / -0

$$\displaystyle \lim _{ \theta \rightarrow \pi /2 }{ \dfrac { 1-\sin \theta }{ (\pi /2-\theta )\cos { \theta } } } $$ is equal to

A
$$1$$

B
$$-1$$

C
$$1/2$$

D
$$-1/2$$

Solution
Question 10
1 / -0

If $$\underset {x \rightarrow \infty}{lim} (1+\frac {a}{x}-\frac {4}{x^{2}})^{2x} =e^{3}$$ , then 'a' is equal to :

A
2

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

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

D
$$\frac {1}{2}$$

Solution