Limits and Continuity Test 56

Result

Limits and Continuity Test 56

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Weekly Quiz Competition

Question 1
1 / -0

$$ \lim _{x \rightarrow0}\left(\frac{e^{x}+e^{-x}-2}{x^{2}}\right)^{1 / x^{2}} $$ is

A
$$

e^{1 / 2}

$$

B
$$

e^{1 / 4}

$$

C
$$

e^{1 / 3}

$$

D
$$

e^{1 / 12}

$$

Solution
Question 2
1 / -0

Let $$a \in \left( 0 , \frac { \pi } { 2 } \right)$$, then the value of
$$ \lim _ { a \rightarrow 0 } \frac { 1 } { a ^ { 3 } } \int _ { 0 } ^ { a } \ell n (1+tan a tan x)dx$$ is equal to

A
$$\frac { 1 }{ 3 } $$

B
$$\frac { 1 }{ 2 } $$

C
$$\frac { 1 }{ 6 } $$

D
1
Question 3
1 / -0

Let x be an irrational, then $$\underset { m\rightarrow \infty }{ lim } \underset { n\rightarrow \infty }{ lim } \left\{ cos(n!\pi x) \right\} ^{ 2m }$$ equals

A
0

B
-1

C
1

D
Indeterminate

Solution
Question 4
1 / -0

$$ \lim _ { x \rightarrow 0 } \frac { \sin x } { x } = y $$

A
$$

y > 1

$$

B
$$

y < 1

$$

C
$$

y \geq 1

$$

D
$$

y \leq 1

$$

Solution
Question 5
1 / -0

$$\underset { \rightarrow }{ Lim } \frac { 1-{ cos }^{ 2 }x }{ xsin2x } $$

A
1/2

B
3/2

C
3/4

D
1/4
Question 6
1 / -0

$$\lim_{x\rightarrow 0}\frac{1-cos^{3}x}{x sib x cos x }$$ is equal to

A
2/5

B
3/5

C
3/2

D
3/4
Question 7
1 / -0

$$\displaystyle x\xrightarrow { lim } 5\quad \left(\frac{\sqrt{1-\cos(2x-10)}}{\sin (x-5)} \right) $$

A
$$-\sqrt{2}$$

B
$$\sqrt{2}$$

C
does not exist

D
none of these

Solution
Question 8
1 / -0

$$\lim _ { x \rightarrow 0 } \int _ { 0 } ^ { x } \dfrac { \left( \tan ^ { - 1 } t \right) ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } d t$$ is equal to

A
$$\pi ^ { 2 }$$

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

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

D
None of these

Solution
Question 9
1 / -0

$$\underset {x\rightarrow 0}{Lim} \frac {sin x}{x} = y $$

A
$$y>1$$

B
$$y<1$$

C
$$y\ge 1$$

D
$$y \le 1$$

Solution
Question 10
1 / -0

$$\underset { x\rightarrow 1 }{ lim } { \left[ cosec { \dfrac { \pi x }{ 2 } } \right] }^{ { 1 }/{ \left( 1-x \right) } }$$ (where $$[.]$$ represents the greatest integer function) is equal to

A
$$0$$

B
$$1$$

C
$$\infty$$

D
$$Does \ not \ exist$$

Solution