Existence of Limit

Q.

${lim}_{x\to 0}\frac{\sqrt{1-cos2x}}{\sqrt{2x}}is$

1. zero

2. Does not exist

3. -1

4. λ

Q.

Let $f\left(x\right)=x{(-1)}^{\left[\frac{1}{x}\right]}.x\ne 0$, where [x] denotes the greatest integer less than or equal to x. then $\underset{x\to 0}{lim}f\left(x\right)$

1. Doesn't exist

2. is equal to 2

3. is equal to 0

4. is equal to -1

Q.

The value of $\underset{x\to 0}{lim}\frac{1n(1+\{x\left\}\right)}{\left\{x\right\}}$ is (where {x} denotes the fractional part of x)

1. Does not exist

2. 1

3. 0

4. 2

Q.

$Letf\left(x\right)=\{\begin{array}{cc}{x}^2& {}_{\frac{k(x^2-4)}{2-x}}\end{array}\text{when x is an int eger otherwise then}{lim}_{x\to 2}f\left(x\right)$

1. exists only when k=1

2. exists for every real k

3. exists for every real k =1

4. does not exist

Q.

$\underset{x\to 0}{lim}\frac{\sqrt{1-cos2x}}{\sqrt{2}x}$ is

(JEE 2002)

1. -1

2. Zero

3. 1

4. Does not exist

Q.

$\underset{x\to 3}{lim}\left(\right[x-3]+[3-x]-x),\text{where[.]denote the greatest integer function, is equal to:}$

1. 4

2. -4

3. 0

4. Does not exist

Q.

The value of $\underset{x\to \frac{\pi }{2}}{lim}\left[si{n}^{-1}sinx\right]$, [x] is the greatest integer function of x, is

1. 0

2. 1

3. 

4. 

Q.

$li{m}_{x\to 1}$ $\frac{1}{|1-x|}$ =

1. 0

2. 1

3. 2

4. 

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin \left[x\right]}{\left[x\right]},& \left[x\right]\ne 0\\ 0,& \left[x\right]=0\end{array},$ where [x] denotes the greatest integer less than or equal to x then $\underset{x\to 0}{lim}f\left(x\right)$ equals

1. 
2. does not exist
3. 
4. 

Q. $f\left(x\right)=\{\begin{array}{cc}4x-3,& x<1\\ x^2& x\ge 1\end{array},$ then
$\underset{x\to 1}{lim}f\left(x\right)=$

1. $-1$
2. $0$
3. $0$
4. $1$