Existence of Limit

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. Does not exist

2. is equal to 2

3. is equal to 0

4. is equal to -1

Q. Which of the following is the principal value branch of ${\tan }^{-1}x$

1. $(0,\pi )$
2. $[-\frac{\pi }{2},\frac{\pi }{2}]$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. $(-\frac{\pi }{2},\frac{\pi }{2})-\left\{0\right\}$

Q. The value of $\underset{x\to 0}{lim}\left[\frac{\sin |x|}{|x|}\right]$ is
(where [.] is greatest integer function)

1. Right hand limit (RHL) is $1$.
2. Left hand limits (LHL) is $-1$.
3. Both LHL and RHL is 0.
4. Both LHL and RHL is $1$.

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.

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

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. 1

2. $\frac{\pi }{2}$

3. 0

4. $\frac{1}{2}$

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

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.

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

(JEE 2002)

1. 1

2. -1

3. Zero

4. Does not exist

Q. $\underset{x\to 1}{lim}\frac{x\sin \left\{x\right\}}{x-1},$ where {x} denotes the fractional part of x, is equal to

1. $-1$
2. $0$
3. $1$
4. does not exist
   

Q.

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

1. $\lambda$

2. -1

3. zero

4. Does not exist

Q. $\underset{x\to 1}{lim}\frac{x\sin \left\{x\right\}}{x-1},$ where {x} denotes the fractional part of x, is equal to

1. $-1$
2. $0$
3. $1$
4. does not exist
   

Q.

consider the function
$f\left(x\right)=\{\begin{array}{ccc}a+bx,x<1& & \\ 4,x=1& & \\ b-ax,x>1& & \end{array}$
If ${lim}_{x\to 1}f\left(x\right)=f\left(1\right)$, then the values of a and b are

1. a = 3, b = 1

2. a = 1, b = 3

3. a = 0, b = 4

4. a = 4, b = 0

Q. Let $f\left(x\right)=(\begin{array}{c}{x}^2-1,\mathrm{if}0<x<2\\ 2x+3,\mathrm{if}2\le x<3\end{array}$, a quadratic equation whose roots are $\underset{x\to 2^{-}}{\lim }f\left(x\right)\mathrm{and}\underset{x\to 2^{+}}{\lim }f\left(x\right)$ is

1. $x^2-6x+9=0$
2. $x^2-7x+8=0$
3. $x^2-14x+49=0$
4. $x^2-10x+21=0$

Q. If f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$ and g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$ is

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