Limits and Continuity Test 37

Result

Limits and Continuity Test 37

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

Consider $$A=\begin{bmatrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{bmatrix}$$, then the value of $$\lim_{n \rightarrow \infty} \dfrac{A^{n}}{n}$$ (where $$\theta \in R$$) is equal to

A
$$10$$

B
zero matrix

C
symmetric matrix

D
$$4$$

Solution
Question 2
1 / -0

If $$\displaystyle\lim_ { x\rightarrow \lambda } { \left( 2-\dfrac { \lambda }{ x } \right) }^{ \lambda tan\left( \dfrac { \pi x }{ 2\lambda } \right) }=\frac { 1 }{ e } ,$$ then $$\lambda $$ is equal to-

A
$$-\pi $$

B
$$\pi $$

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

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

Solution
Question 3
1 / -0

$$\displaystyle \lim_{x \rightarrow 0}\dfrac {1}{x\sqrt {x}}\left(a\ arc\ tan \dfrac {\sqrt {x}}{a}-b\ arc\ \tan \dfrac {\sqrt {x}}{b}\right)$$ has the value equal to

A
$$\dfrac {a-b}{3}$$

B
$$0$$

C
$$\dfrac {(a^{2}-b^{2})}{6a^{2}b^{2}}$$

D
$$\dfrac {a^{2}-b^{2}}{3a^{2}b^{2}}$$

Solution
Question 4
1 / -0

For each $$t\in R$$, let [t] be the greatest integer less than or equal to t. Then,
$$\underset { { x\rightarrow 0 }^{ + } }{ lim } x([\frac { 1 }{ x } ]+[\frac { 2 }{ x } ]+...+[\frac { 15 }{ x } ])$$

A
is equal to 0

B
is equal to 15

C
is equal to 120

D
does not exist (in R)

Solution
Question 5
1 / -0

$$\displaystyle \lim _{ x\rightarrow 2 }{ \frac { \sqrt [ 3 ]{ 60+{ x }^{ 2 } } -4 }{ \sin { \left( x-2 \right) } } } $$

A
$$\dfrac {1}{4}$$

B
$$0$$

C
$$\dfrac {1}{12}$$

D
$$Does\ not\ exist$$

Solution
Question 6
1 / -0

The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ \csc^{ 4 }{ x } \int _{ 0 }^{ { x }^{ 2 } }{ \frac { ln\left( 1+4t \right) }{ { t }^{ 2 }+1 } } dt } $$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 7
1 / -0

The value of $$\displaystyle\lim_{n\rightarrow \infty}n(n\{ln (n)-ln (n+1)\}+1)$$ is?

A
e

B
$$\dfrac{1}{e}$$

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

D
$$\dfrac{1}{4}$$

Solution
Question 8
1 / -0

$$\displaystyle \lim_{x\rightarrow \infty}{x^{2}\sin\left(\log_{e}\sqrt{\cos\dfrac{\pi}{x}}\right)}$$

A
$$0$$

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

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

D
$$-\dfrac{\pi^{2}}{8}$$

Solution
Question 9
1 / -0

$$\displaystyle \lim _{ x\rightarrow \frac { \pi }{ 4 } }{ { \left( \sin { 2x } \right) }^{ \sec ^{ 2 }{ 2x } } }$$ is equal to

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

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

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

D
$$e^{\dfrac {1}{2}}$$

Solution
Question 10
1 / -0

$$\underset { x\rightarrow \frac { \pi }{ 2 } }{ lim } \frac { (1-sinx)({ 8x }^{ 2 }-{ \pi }^{ 3 })cosx }{ { (\pi -2x) }^{ 4 } } $$

A
$$-\cfrac { { \pi }^{ 2 } }{ 16 } $$

B
$$\cfrac { { 3\pi }^{ 2 } }{ 16 } $$

C
$$\cfrac { { \pi }^{ 2 } }{ 16 } \quad $$

D
$$-\cfrac { 3{ \pi }^{ 2 } }{ 16 } \quad $$

Solution