Limits and Continuity Test 58

Result

Limits and Continuity Test 58

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Question 1
1 / -0

f(X)=|x|+|x-1| is continuous at

A
'0' only

B
0,1 only

C
Every where

D
No where

Solution

$$\textbf{Step-1: Finding values of}$$ $$\mathbf{f(x)}$$
$$f(x)=|x|+|x-1|$$
$$f(x)=-x-|x-1|$$ $$\text{at}$$ $$x\leq{0}$$
$$=x-(x-1)$$ $$\text{at}$$ $$0\leq{x}<1$$
$$=x+(x-1)$$ $$\text{at}$$ $$x\geq{1}$$
$$\text{Hence, we get}$$
$$f(x)=1-2x$$ $$\text{at}$$ $$x\leq{0}$$
$$=1$$ $$\text{at}$$ $$0\leq{x}<1$$
$$=2x-1$$ $$\text{at}$$ $$x\geq{1}$$
$$\textbf{Step-2: Finding limits of the function at constraint values}$$
$$\text{At}$$ $$x=0,$$
$$\lim_{x\to 0} f(x)=1$$
$$\text{Hence,}$$ $$f(x)$$ $$\text{is continuous at}$$ $$x=0$$
$$\text{At}$$ $$x=1,$$
$$\lim_{x\to 1} f(x)=1$$
$$\text{Hence,}$$ $$f(x)$$ $$\text{is continuous at}$$ $$x=1$$
$$\text{Hence,}$$ $$f(x)$$ $$\text{is continuous everywhere}$$

$$\textbf{Hence,Correct option is (C)}$$
Question 2
1 / -0

The value of $$\underset{x\rightarrow 1}{lim}(2-x)^{tan\left(\dfrac{\pi x}{2}\right)}$$ is

A
$$e^{-2/pi}$$

B
$$e^{1/pi}$$

C
$$e^{2/pi}$$

D
$$e^{-1/pi}$$

Solution
Question 3
1 / -0

$$\displaystyle \lim_{x\rightarrow 0}\dfrac {\sin x - x}{x^{3}}$$ is equal to

A
$$-\dfrac {1}{6}$$

B
$$-\dfrac {1}{3}$$

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

D
$$-1$$

Solution
Question 4
1 / -0

$$\underset{x \rightarrow 1} {lim}\dfrac{x^2-1}{\sin^2x+\cos x\cos (x+2)-\cos^2(x+1)}$$ is-

A
$$0$$

B
$$\dfrac{1}{\cos 1}$$

C
$$\dfrac{2}{\sin 2}$$

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

Solution
Question 5
1 / -0

$$\lim_{x\rightarrow 1}\frac{1-x^{-2/3}}{1-x^{-1/3}}$$

A
2

B
1

C
0

D
does not exist
Question 6
1 / -0

$$\overset {lim}{x \rightarrow \pi/2} \dfrac{\sin(x \ cos x)}{cos(x\, \ sin x)}$$ is equal to

A
$$1$$

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

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

D
does not exist

Solution
Question 7
1 / -0

$$\displaystyle \lim _{x \rightarrow 1}\left(\dfrac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\dfrac{1-\cos (x+1)}{(x+1)^{2}}} $$ is equal to:

A
1

B
$$(2 / 3)^{1 / 2} $$

C
$$ (3 / 2)^{1 / 2} $$

D
$$ e^{1 / 2} $$

Solution
Question 8
1 / -0

$$ \displaystyle \lim _{x \rightarrow 0}\left(\dfrac{1^{x}+2^{x}+3^{x}+\cdots+n^{x}}{n}\right)^{1 / x} $$ is equal to

A
$$ (n !)^{n} $$

B
$$ (n !)^{1 / n} $$

C
$$ n ! $$

D
$$ \ln (n !) $$

Solution
Question 9
1 / -0

If $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x} $$ is non-zero finite, then $$ n $$ must be equal

A
4

B
1

C
2

D
3

Solution
Question 10
1 / -0

$$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}} $$ is equal to

A
0

B
1

C
-1

D
Does not exists

Solution