Limits and Continuity Test 47

Result

Limits and Continuity Test 47

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle\lim_{x\rightarrow 0}\dfrac{x\tan 2x-2x\tan x}{(1-\cos 2x)^2}$$ is equal to?

A
$$2$$

B
$$-2$$

C
$$1/2$$

D
$$-1/2$$

Solution
Question 2
1 / -0

If $$f(x)$$ is odd linear polynomial with $$f(1)=1$$, then $$\underset{x \to 0} {\lim} \dfrac{2^{f(\tan x)}-2^{f(\sin x)}}{x^{2}f(\sin x)}$$ is

A
$$1$$

B
$$\ln 2$$

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

D
$$\cos 2$$

Solution
Question 3
1 / -0

If $$\displaystyle\lim_{x\rightarrow 0}[1+x ln (1+b^2)]^{1/x}=2b\sin^2\theta, b > $$ s m f $$\theta\in (-\pi, \pi]$$, then the value of $$\theta$$ is?

A
$$\pm \dfrac{\pi}{4}$$

B
$$\pm \dfrac{\pi}{3}$$

C
$$\pm \dfrac{\pi}{6}$$

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

Solution
Question 4
1 / -0

$$\lim _ { x \rightarrow \infty } \dfrac { \sin \left( \pi \cos ^ { -2 } x \right) } { x ^ { 2 } }$$ is equal to:

A
$$- \pi$$

B
$$ \pi$$

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

D
$$1$$

Solution
Question 5
1 / -0

The value of $$\underset { x\rightarrow 0 }{ lim } \frac { { 27 }^{ x }-{ 9 }^{ x }-{ 3 }^{ x }+1 }{ \sqrt { 5 } -\sqrt { 4+cosx } } $$ is

A
$$\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$

B
$$8\sqrt { 5 } log3$$

C
$$16\sqrt { 5 } log3$$

D
$$8\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$

Solution
Question 6
1 / -0

$$\lim _ { x \rightarrow 0 } \dfrac { | \cos ( \sin ( 3 x ) ) | - 1 } { x ^ { 2 } }$$ equals

A
$$\dfrac { - 9 } { 2 }$$

B
$$\dfrac { - 3 } { 2 }$$

C
$$\dfrac { 3 } { 2 }$$

D
$$\dfrac { 9 } { 2 }$$

Solution
Question 7
1 / -0

$$\lim _ { x \rightarrow 0 } \left( \left[ \dfrac { - 5 \sin x } { x } \right] + \left[ \dfrac { 6 \sin x } { x } \right] \right)$$ (where $$[ .]$$ denotes greatest integer function) is equal to

A
$$0$$

B
$$-12$$

C
$$1$$

D
$$2$$

Solution
Question 8
1 / -0

$$\displaystyle\lim_{x\rightarrow \dfrac{\pi}{4}}\dfrac{\cos x-\sin x}{\left(\dfrac{\pi}{4}-x\right)(\cos x+\sin x)}=?$$

A
$$2$$

B
$$1$$

C
$$0$$

D
$$3$$

Solution
Question 9
1 / -0

$$\displaystyle\lim_{x\rightarrow 0}\left\{\dfrac{log_e(1+x)}{x^2}+\dfrac{x-1}{x}\right\}$$ is equal to?

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

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

C
$$1$$

D
None of these

Solution
Question 10
1 / -0

Consider $$\lim _ { x \rightarrow 0 } \dfrac { a x + b e ^ { - x } + \sin x + 1 } { a x - b \sin x } = \ell ( \ell$$ is a finite number)

A
$$\ell = 0 , \text { if } a = - 2 , b = - 1$$

B
$$\ell = 5 , i f a = 1 , b = - 1$$

C
$$\ell = 1 , \text { if } a = b = - 1$$

D
None of these

Solution