Limits and Continuity Test 46

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Limits and Continuity Test 46

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { { \left( \cos { x } \right) }^{ 1/2 }-{ \left( \cos { x } \right) }^{ 1/3 } }{ \sin ^{ 2 }{ x } } } $$ is $$

A
$$1/6$$

B
$$-1/12$$

C
$$2/3$$

D
$$1/3$$

Solution
Question 2
1 / -0

$$\displaystyle \lim _{ x\rightarrow 0 }{ \frac { x\left( { e }^{ \sin { x } }-1 \right) }{ 1-\cos { x } } } $$

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

B
$$2$$

C
$$0$$

D
$$1$$

Solution
Question 3
1 / -0

$$\underset { x\rightarrow 0 }{ lim } (\cos x+a\sin b{ x) }^{ \frac { 1 }{ x } }$$ is equal to

A
$$e^a$$

B
$$e^{ab}$$

C
$$e^b$$

D
$$e^{a/b}$$

Solution
Question 4
1 / -0

$$\displaystyle \lim_{x\rightarrow 0}{\dfrac{x(1+a\cos x)-b\sin x}{x^{3}}}=1$$ then

A
$$a=-5/2$$

B
$$a=-3/2,b=-1/2$$

C
$$a=-3/2,b=-5/2$$

D
$$a=-5/2,b=-3/2$$

Solution
Question 5
1 / -0

The value of $$\displaystyle\lim_{\theta \rightarrow 0^+}\dfrac{\sin\sqrt{\theta}}{\sqrt{\sin\theta}}$$ is equal to?

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$4$$

Solution
Question 6
1 / -0

$$\displaystyle\lim_{x\rightarrow 0}\left(\dfrac{1}{\sin^2x}-\dfrac{1}{\sin h^2x}\right)=?$$

A
$$2_{\displaystyle /3}$$

B
$$0$$

C
$$1_{\displaystyle /3}$$

D
$$-2_{\displaystyle /3}$$

Solution
Question 7
1 / -0

Let $$f$$ : $$\left ( \dfrac{\pi}{2}, \dfrac{\pi}{2} \right )\rightarrow R$$ , $$f(x) = \left \{\begin{matrix} \lim_{n\rightarrow \infty }\dfrac{(tanx)^{2n} + x^2}{sin^2x + (tanx)^{2n}}; & x \neq 0 \\ 1; & x = 0 \end{matrix} \right., n \in N$$ . Which of the following holds good ?

A
$$f \left( -\dfrac{\pi^-}{4} \right) = f \left (\dfrac{\pi^+}{4} \right)$$

B
$$f \left( -\dfrac{\pi^-}{4} \right) = f \left (-\dfrac{\pi^+}{4} \right)$$

C
$$f \left( \dfrac{\pi^-}{4} \right) = f \left (\dfrac{\pi^+}{4} \right)$$

D
$$f(0^+) = f(0) = f(0^)$$
Question 8
1 / -0

If $$ \lim _{x \rightarrow 0}\left(\cos x+a^{3} \sin \left(b^{6} x\right)\right)^{\frac{1}{x}}=e^{512} $$ then value of $$ab^2$$ is equal to

A
$$-512$$

B
$$512$$

C
$$8$$

D
none of these

Solution
Question 9
1 / -0

$$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sin(\pi \cos^2x)}{x^2}$$ is equal to?

A
$$-\pi$$

B
$$\pi$$

C
$$\pi/2$$

D
$$1$$

Solution
Question 10
1 / -0

The value of $$\begin{matrix} lim \\ x\rightarrow y \end{matrix}\dfrac { { sin }^{ 2 }x-sin^{ 2 }y }{ { x }^{ 2 }-{ y }^{ 2 } } $$

A
0

B
1

C
$$\dfrac { siny }{ y } $$

D
$$\dfrac { sin2y }{ 2y } $$