Limits and Continuity Test 36

Result

Limits and Continuity Test 36

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Question 1
1 / -0

$$\displaystyle\underset{x\rightarrow 0}{Lt}\left(cosec x-\dfrac{1}{x}\right)=?$$

A
$$0$$

B
$$1/2$$

C
$$1$$

D
Does not exits

Solution
Question 2
1 / -0

If $$f(x)=\begin{cases} \dfrac { 1-\sqrt { 2 } \sin { x } }{ \pi -4x } ,\quad \quad ifx\neq \dfrac { \pi }{ 4 } \\ a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\quad \quad ifx=\dfrac { \pi }{ 4 } \end{cases}$$ is continous at $$x=\dfrac {\pi}{4}$$ then $$a=$$

A
$$4$$

B
$$2$$

C
$$1$$

D
$$1/4$$

Solution
Question 3
1 / -0

$$If {A_i} = \frac{{x - {a_i}}}{{\left| {x - {a_i}} \right|}}, \,i = 1,2,3,.....n$$ and $${a_1}< {a_2}< {a_3}....< {a_{n,}} \, then$$
$$\mathop {\lim }\limits_{x \to {a_m}} \left( {{A_1}{A_2}......{A_n}} \right), 1 \le m \le n$$

A
is equal to $${\left( { - 1} \right)^m}$$

B
is equal to $${\left( { - 1} \right)^{m + 1}}$$

C
is equal to $${\left( { - 1} \right)^{n-m}}$$

D
is equal to $${\left( { - 1} \right)^{n-m-1}}$$

Solution
Question 4
1 / -0

$$\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\frac{\pi }{2} - x} \right)}^3}}}$$

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

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

C
$$2$$

D
$$1$$

Solution
Question 5
1 / -0

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { \sin ^{ -1 }{ x } -\tan ^{ -1 }{ x } }{ { x }^{ 3 } } } $$ is equal to

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$\dfrac{1}{2}$$

Solution
Question 6
1 / -0

If $$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + x + 1}}{{x + 1}} - ax - b} \right)\, = 4$$,then

A
$$a=1,b=4$$

B
$$a=1,b=-4$$

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

D
$$a=2,b=3$$

Solution
Question 7
1 / -0

$$f(x)= x\sin\dfrac{1}{x} , \ for x\neq 0$$
$$= 0,\ for x=0$$
Then.

A
$$f'(0^+) exit\ but \ f'(0^-)$$ does not exit

B
$$f'(0^+) \ and \ f'(0^-)$$ do not exit

C
$$f'(0^+) = f'(0^-)$$

D
none of these

Solution
Question 8
1 / -0

$$\displaystyle\lim_{n\rightarrow \infty}\left(\tan\theta +\dfrac{1}{2}\tan \dfrac{\theta}{2}+\dfrac{1}{2^2}\tan \dfrac{\theta}{2^2}+...+\dfrac{1}{2^n}\tan\dfrac{\theta}{2^n}\right)$$ equals?

A
$$\dfrac{1}{\theta}$$

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

C
$$2\cot 2\theta$$

D
None of these

Solution
Question 9
1 / -0

Let p= $$\lim_{x\rightarrow 0+}(1+tan^{2}\sqrt{x})^{\frac{1}{2x}}$$ then log p is equal to :

A
1

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

C
$$\frac{1}{4}$$

D
2

Solution
Question 10
1 / -0

$$f(x) = \left\{\begin{matrix}(3/x^{2})\sin 2x^{2} & if x M 0 \\\dfrac {x^{2} + 2x + c}{1 - 3x^{2}} & if\ x \geq 0, x \neq \dfrac {1}{\sqrt {3}}\\ 0 & x = 1/ \sqrt {3}\end{matrix}\right.$$ then in order that $$f$$ be continuous at $$x = 0$$, the value of $$c$$ is

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution