Limits and Continuity Test 2

Result

Limits and Continuity Test 2

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

Use limit properties to evaluate $$\displaystyle\lim_{x\to4}\dfrac{3x^2\tan \dfrac {\pi}{x}}x $$

A
$$12$$

B
$$14$$

C
$$16$$

D
$$18$$

Solution

$$\displaystyle\lim_{x\to4}\dfrac{3x^2\tan \dfrac {\pi}{x}}x =\lim_{x\to 4}\left(3x\tan\dfrac{\pi}{x}\right)$$, simplify

$$=3(4)\tan\dfrac{\pi}{4}=12(1)=12$$, [substitute the limit ]

The from of the limit is not indeterminate, so we can substitute the limit value
Question 2
1 / -0

Evaluate $$\underset{x \rightarrow 3}\lim \sqrt[4] {x^3}$$ using the properties of limits.

A
$$28^{1/4}$$

B
$$25^{1/4}$$

C
$$27^{1/4}$$

D
$$26^{1/4}$$

Solution

$$\displaystyle \lim _{ x\rightarrow 3 }{ \sqrt [ 4 ]{ { x }^{ 3 } } } \\={ (3^3) }^{ { 1 }/{ 4 } }={ (27) }^{ { 1 }/{ 4 } }$$
Question 3
1 / -0

$$\underset{x \rightarrow 0}{lim}(1+ax)^{\large{\frac{b}{x}}}$$ is equal to

A
$$e^{ab}$$

B
$$e^{a+b}$$

C
non-existent

D
none of these

Solution

Now,
$$\underset{x \rightarrow 0}{lim}(1+ax)^{\large{\frac{b}{x}}}$$
$$=\underset{x \rightarrow 0}{lim}\left\{(1+ax)^{\large{\frac{1}{ax}}}\right\}^{ab}$$
$$=\left\{\underset{x \rightarrow 0}{lim}(1+ax)^{\large{\frac{1}{ax}}}\right\}^{ab}$$
$$=e^{ab}$$. [ Since $$\lim\limits_{x \to 0}(1+x)^{\large{\frac{1}{x}}}=e$$].
Question 4
1 / -0

The integer $$'n'$$ for which $$\mathop {\lim }\limits_{x \to 0} \dfrac{{\cos 2x - 1}}{{{x^n}}}$$ is a finite non-zero number is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

$$\begin{array}{l}\mathop {\lim }\limits_{x \to 0} \cfrac{{\cos 2x - 1}}{{{x^n}}} = \mathop {\lim }\limits_{x \to 0} \cfrac{{ - 2{{\sin }^2}x}}{{{x^n}}}\\at,n = 2\\\mathop {\lim }\limits_{x \to 0} \cfrac{{ - 2{{\sin }^2}x}}{{{x^2}}} = - 2\end{array}$$
which is a finite non-zero number
Question 5
1 / -0

Solve: $$\displaystyle \underset{n\rightarrow \infty}{\lim}\pi^3 \left(1-\dfrac{4}{n^2}\right)$$

A
$$\pi^3-1$$

B
$$\pi^3$$

C
$$\pi^3+1$$

D
$$-3\pi^3$$

Solution

$$\displaystyle \lim _{ n\rightarrow \infty }{ { \pi }^{ 3 } } \left(1-\frac { 4 }{ { n }^{ 2 } } \right)$$
$$={ \pi }^{ 3 }(1-0)\\ ={ \pi }^{ 3 }$$
Question 6
1 / -0

If a sequence $$< a_{n} >$$ is such that $$a_{1},a_{n+1}=\dfrac {2+3a_{n}}{1+2a_{n}}$$ and $$\displaystyle \lim_{n \rightarrow \infty}a_{n}$$ exists, then $$a_{n}$$ is equal to

A
$$\cos 36^{o}$$

B
$$2\cos 36^{o}$$

C
$$\sin 18^{o}$$

D
$$2\sin 36^{o}$$
Question 7
1 / -0

$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$ equals

A
$$\dfrac{7}{3}$$

B
$$\dfrac{10}{3}$$

C
$$\dfrac{14}{3}$$

D
$$\dfrac{1}{3}$$

Solution

We have,
$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\dfrac{\sin7x}{7x}\times 7x}{\dfrac{\sin3x}{3x}\times 3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\dfrac{\sin7x}{7x}\times 7}{\dfrac{\sin3x}{3x}\times 3}$$
$$=\dfrac{1\times 7}{1\times 3}$$
$$=\dfrac{7}{3}$$

Hence, this is the answer.
Question 8
1 / -0

$$\displaystyle \lim_{x\rightarrow 0}{(1+\sin x)^{\cos x}}$$ is equal to

A
$$0$$

B
$$e$$

C
$$1$$

D
$$\dfrac{1}{e}$$

Solution

$$\displaystyle{\lim}_{x\rightarrow 0}{\left(1+\sin{x}\right)}^{\cos{x}}$$
$$={\left(1+\sin{0}\right)}^{\cos{0}}$$
$$={1}^{1}=1$$
'
Question 9
1 / -0

Evaluate the following limit :
$$lim_{x\rightarrow 0} \dfrac{1-\cos 2x}{x^2}$$

A
$$0$$

B
$$1$$

C
$$2$$

D
none of these

Solution

$$lim_{x\rightarrow 0} \dfrac{1-\cos 2x}{x^2}$$

$$=lim_{x\rightarrow 0} \dfrac{2\sin^2 x}{x^2}$$

$$=2lim_{x \rightarrow 0} (\dfrac{\sin x}{x} \times \dfrac{\sin x}{x})$$

$$=2lim_{x\rightarrow 0} \dfrac{\sin x}{x} \times lim_{x\rightarrow 0} \dfrac{\sin x}{x}=2(1)(1)=2$$
Question 10
1 / -0

Evaluate $$\displaystyle \lim_{n\rightarrow \infty} \dfrac{1+2+3+...+n}{n^2}$$

A
$$\cfrac { 1 }{ 2 } $$

B
$$1$$

C
$${ 3 }^{ 2 }$$

D
$$\cfrac { 1 }{ { 2 }^{ 2 } } $$

Solution

$$\displaystyle \lim_{n\rightarrow \infty} \dfrac{1+2+3+...+n}{n^2}$$

$$\displaystyle =\lim_{n\rightarrow \infty} \dfrac{1}{n^2} \times \dfrac{n(n+1)}{2}$$

$$\displaystyle =\lim_{n\rightarrow \infty} \dfrac{1}{2}(1+\dfrac{1}{n})=\dfrac{1}{2}$$

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Limits and Continuity Test 2

Result

Limits and Continuity Test 2

Score

-

Rank

-

SHARING IS CARING

Question 1

$$12$$

$$14$$

$$16$$

$$18$$

Solution

Question 2

$$28^{1/4}$$

$$25^{1/4}$$

$$27^{1/4}$$

$$26^{1/4}$$

Solution

Question 3

$$e^{ab}$$

$$e^{a+b}$$

non-existent

none of these

Solution

Question 4

$$1$$

$$2$$

$$3$$

$$4$$

Solution

Question 5

$$\pi^3-1$$

$$\pi^3$$

$$\pi^3+1$$

$$-3\pi^3$$

Solution

Question 6

$$\cos 36^{o}$$

$$2\cos 36^{o}$$

$$\sin 18^{o}$$

$$2\sin 36^{o}$$

Question 7

$$\dfrac{7}{3}$$

$$\dfrac{10}{3}$$

$$\dfrac{14}{3}$$

$$\dfrac{1}{3}$$

Solution

Question 8

$$0$$

$$e$$

$$1$$

$$\dfrac{1}{e}$$

Solution

Question 9

$$0$$

$$1$$

$$2$$

none of these

Solution

Question 10

$$\cfrac { 1 }{ 2 } $$

$$1$$

$${ 3 }^{ 2 }$$

$$\cfrac { 1 }{ { 2 }^{ 2 } } $$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.