Limits and Continuity Test 54

Result

Limits and Continuity Test 54

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Weekly Quiz Competition

Question 1
1 / -0

If $$f\left( x \right) =\sqrt { 1-\sqrt { 1-{ x }^{ 2 } } } $$, then $$f(x)$$ is

A
continuous on$$[-1,1]$$

B
differentiable on $$(-1,0)\cup (0,1)$$

C
both (a) and (b)

D
None of the above

Solution
Question 2
1 / -0

$$\mathop {\lim }\limits_{x \to \pi /2} \left[ {x\tan x - \left( {\frac{\pi }{2}} \right)\sec x} \right]$$ is equal to

A
1

B
-1

C
0

D
None of these

Solution
Question 3
1 / -0

$$\underset { x\rightarrow 0 }{ lim } \cfrac { 8 }{ { x }^{ 8 } } \left( 1-cos\cfrac { { x }^{ 2 } }{ 2 } -cos\cfrac { { x }^{ 2 } }{ 4 } +cos\cfrac { { x }^{ 2 } }{ 2 } .cos\cfrac { { x }^{ 2 } }{ 4 } \right) =$$

A
$$\cfrac { 1 }{ 16 } $$

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

C
$$\cfrac { 1 }{ 32 } $$

D
$$1$$

Solution
Question 4
1 / -0

The value of $$\lim _ { x \rightarrow \dfrac { 1 } { 2 } } \dfrac { 2 \sin ^ { - 1 } x - \dfrac { \pi } { 2 } } { 1 - 2 x ^ { 2 } }$$ is equal to

A
$$-1$$

B
$$0$$

C
$$1$$

D
$$2$$

Solution
Question 5
1 / -0

$$f(x) = \log_{1 - 2x}(1 + 2x)$$ for $$ x \ne 0$$
$$= k$$ for $$x = 0$$
is continuous at $$x = 0$$, find $$k.$$

A
$$1$$

B
$$-1$$

C
$$0$$

D
$$\frac{1}{2}$$
Question 6
1 / -0

If $$ \alpha , \beta , \ in (-\frac{\pi}{2},0) $$ such that $$(sin \alpha +sin \beta ) +\frac{sin \alpha }{sin \beta} =0 $$ and $$(sin \alpha +sin \beta ) \frac{sin \alpha}{sin \beta }=-1 $$ and $$\lambda =\begin{matrix} lim \\ n\rightarrow \infty \end{matrix}\frac { 1+(2sin\quad theta\quad ){ }^{ 2n } }{ (2sin\quad \quad beta\quad ){ }^{ 2n } } $$ then

A
$$2 \alpha +3 \beta = \frac{-5 \pi }{6}$$

B
$$ \lambda \pi +\alpha +\beta =\frac{5 \pi}{6}$$

C
$$ \alpha - \beta = \frac{ \pi }{3}$$

D
$$ \alpha + \beta = \frac{- \pi }{3}$$

Solution
Question 7
1 / -0

$$\mathop {\lim }\limits_{x \to \infty } \left( {\dfrac{{{x^2}\sin \left( {\dfrac{1}{x}} \right) - x}}{{1 - \left| x \right|}}} \right) = $$

A
0

B
1

C
-1`

D
2

Solution
Question 8
1 / -0

$$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}(cos\quad +\quad sinx{ ) }^{ 1/x }$$ is equal to

A
e

B
$${ e }^{ 2 }$$

C
$${ e }^{ -1 }$$

D
1

Solution
Question 9
1 / -0

$$ \lim _{x \rightarrow a}\left(2-\frac{a}{x}\right)^{\tan \left(\frac{\pi x}{2 a}\right)} $$

A
$$

e^{-\frac{a}{\pi}}

$$

B
$$

e^{-\frac{2 a}{\pi}}

$$

C
$$

e^{-\frac{2}{x}}

$$

D
1

Solution
Question 10
1 / -0

$$\lim _{ x\rightarrow 0 }{ \frac { \sin { [\cos { x } ] } }{ 1+[\cos { x } ] } } $$ is (where [] is G.I.F)

A
1

B
0

C
does not exist

D
2

Solution