Limits and Continuity Test 43

Result

Limits and Continuity Test 43

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Question 1
1 / -0

$$\displaystyle \lim _{ \theta \rightarrow 0 }{ \frac { 4\theta \left( \tan { \theta -2\theta \tan { \theta } } \right) }{ { \left( 1-\cos { 2\theta } \right) }^{ 2 } } } $$ is

A
$$1/\sqrt {2}$$

B
$$1/2$$

C
$$1$$

D
$$2$$

Solution
Question 2
1 / -0

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { 3\sin { \left( { x }^{ 9} \right) -\sin { \left( { x }^{ 9 } \right) } } }{ { x }^{ 3 } } } =q$$

A
0

B
$$4\left(\dfrac{\pi}{200}\right)^{3}$$

C
$$\dfrac{\pi}{200}$$

D
$$\dfrac{\pi}{100}$$

Solution
Question 3
1 / -0

Let $$f(x)=(1+\sin x)^{\csc x}$$, the value of $$f(0)$$ so that $$f$$ is a continuous function is

A
$$e$$

B
$$e^{1/2}$$

C
$$e^{2}$$

D
$$1$$

Solution

$$\begin{array}{l} If\, \, m\left( x \right) \, \, and\, \, g\left( x \right) \, are\, \, two\, functions\, \, such\, \, that \\ { \lim }_{ x\to a }m\left( x \right) =0={ \lim }_{ x\to a }g\left( x \right) \\ now, \\ { \lim }_{ x\to a }\left( { 1+m{ { \left( x \right) }^{ \frac { 1 }{ { g\left( x \right) } } } } } \right) ={ \lim }_{ x\to a }\frac { { m\left( x \right) } }{ { g\left( x \right) } } \\ \to it\, is\, useful\, \, fro\, \, \left( { { 1^{ \infty } } } \right) \, \, from \\ here,\, \, m\left( x \right) =\sin x\, \, g\left( x \right) \, =\sin x \\ f\left( 0 \right) =\, \, ={ \lim }_{ x\to a }{ \left( { 1+\sin x } \right) ^{ 1/\sin x } } \\ =e={ \lim }_{ x\to a }\frac { { m\left( x \right) } }{ { g\left( x \right) } } \frac { { \sin x } }{ { \sin x } } \\ ={ e^{ 1 } }=e \end{array}$$
Question 4
1 / -0

If $$k$$ is an integer such that $$\lim_{n \rightarrow \infty} \left[\left(\cos \dfrac{k\pi}{4}\right)^{2}-\left(\cos \dfrac{k\pi}{6}\right)^{2}\right]=0$$ then :

A
$$k$$ is divisible neither by $$4$$ nor by $$8$$.

B
$$k$$ must be divisible by $$12$$ but not necessarily by $$24$$.

C
$$k$$ must be divisible by $$24$$.

D
either $$k$$ is divisible by $$24$$ or $$k$$ is neither divisible by $$4$$ nor by $$6$$.

Solution
Question 5
1 / -0

The value of $$\underset{x \rightarrow 0}{lim} \cos \,ec^4 \,x \displaystyle \overset{x^2}{\underset{0}{\int}} \dfrac{In(1 + 4t)}{1 + t^2}dt$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 6
1 / -0

$$\lim _ { x \rightarrow 0 } \left\{ \tan \left( \frac { \pi } { 4 } + x \right) \right\} ^ { \frac { 1 } { x } } =$$

A
$$e$$

B
$$e ^ { 2 }$$

C
$$e ^ { - 1 }$$

D
$$e ^ { - 2 }$$

Solution
Question 7
1 / -0

If $$g(x)=\frac { x }{ \left[ x \right] } for\quad x>2\quad then\quad \underset { x\rightarrow { 2 }^{ + } }{ Lim } \frac { g\left( x \right) -g\left( 2 \right) }{ x-2 } $$

A
-1

B
0

C
1/2

D
1

Solution
Question 8
1 / -0

$$\lim_{n \rightarrow \infty}n^{2}\left(x^{1/n}-x^{1/\left(n+n\right)}\right),x>0$$, is equal to

A
$$0$$

B
$$e^{x}$$

C
$$\log_{e}x$$

D
$$none\ of\ these$$

Solution
Question 9
1 / -0

$$\lim_{x\rightarrow 0}\frac{ln(sin 3x)}{ln(sin x)}$$ is equal to

A
0

B
1

C
2

D
Non existent

Solution
Question 10
1 / -0

$$\underset{x \rightarrow 2}{lim} \dfrac{\sqrt[3]{60 + x^2} - 4}{\sin (x - 2)}$$ equals

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

B
$$0$$

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

D
Does not exist

Solution