Non Removable Discontinuities

Q.

$Find\underset{x\to 0}{lim}+sgn\left(x\right)+\underset{x\to 0}{lim}-sgn\left(x\right),wheresgn\left(x\right)$ represents the signum function

___

Q. Let $R$ be the set of real numbers and $f:R\to R$ be defined by $f\left(x\right)=\frac{\left\{x\right\}}{1+[x{]}^2},$ where $\left[x\right]$ is the greatest integer less than or equal to $x,$ and $\left\{x\right\}=x-\left[x\right].$ Which of the following statements are true?

I. The range of $f$ is a closed interval.
II. $f$ is continuous on $R.$
III. $f$ is one-one on $R$

1. I only
2. II only
3. III only
4. None of I, II and III

Q. Find $\underset{x\to 5}{lim}f\left(x\right),$ where $f\left(x\right)=|x|-5$

Q. If $f\left(x\right)=\{\begin{array}{cc}7-4x,& x<1\\ x^2+2,& x\ge 1\end{array}$, then the value of $\underset{x\to 1}{lim}f\left(x\right)$ is

1. $3$
2. $2$
3. Does not exist
4. $1$

Q. Let $f\left(x\right)$ be a function continuous $\forall x\in R-\left\{0\right\}$ such that $f^{\prime }\left(x\right)<0,\forall x\in (-\infty ,0)$ and $f^{\prime }\left(x\right)>0,\forall x\in (0,\infty ).$ If $\underset{x\to 0^{+}}{lim}f\left(x\right)=3,\underset{x\to 0^{-}}{lim}f\left(x\right)=4$ and $f\left(0\right)=5,$ then the image of the point $(0,1)$ about the line, $y\cdot \underset{x\to 0}{lim}f({\cos }^{3}x-{\cos }^{2}x)=x\cdot \underset{x\to 0}{lim}f({\sin }^{2}x-{\sin }^{3}x),$ is

1. $(\frac{12}{25},\frac{9}{25})$
2. $(\frac{12}{25},\frac{-9}{25})$
3. $(\frac{16}{25},\frac{-8}{25})$
4. $(\frac{24}{25},\frac{-7}{25})$

Q. Find all the points of discontinuity of the greatest integer function defined by $f\left(x\right)=\left[\frac{x}{3}\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Q. $3x-1\le 5$ for $x\in W$

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

Q.

The given graph is discontinuous at $x=a$ because the limit doesn't exist at $x=a$

1. True

2. False

Q. Let $f\left(x\right)=\{\begin{array}{c}1+|x|,x<-1\\ \left[x\right],x\ge -1\end{array}$ where $[\cdot ]$ denotes the greatest integer function. Then $f\left\{f\right(-2.3\left)\right\}$ is equal to

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

Q. If $f\left(x\right)=\left[x\right]+[-x],$ then which of the following option is correct?
(where $[.]$ represents greatest integer function)

1. $f\left(x\right)$ has missing point discontinuity.
2. $f\left(x\right)$ has isolated point discontinuity.
3. $f\left(x\right)$ has finite type discontinuity.
4. $f\left(x\right)$ has infinite type discontinuity.

Q.

Let $f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$ where $[\cdot ]$ denotes the greatest integer function, then which of the following is true?

1. f(x) has a removable discontinuity at x=1

2. f(x) has a non- removable discontinuity at x=2.

3. f(x) is discontinuous at all positive integers.

4. None of these

Q. If $f\left(x\right)=\{\begin{array}{ccc}\frac{x}{e^{\frac{1}{x}}+1},& when& x\ne 0\\ 0,& when& x=0\end{array}$ then

1. f(x) is continuous at x = 0
2. None of these
3. ${lim}_{x\to 0+}f\left(x\right)=1$
4. ${lim}_{x\to 0-}f\left(x\right)=1$

Q.

$li{m}_{x\to 0}\left(\frac{3x+|x|}{7x-5|x|}\right)$=___

1. $\frac{1}{6}$

2. $\frac{3}{7}$

3. 2

4. does not exist

Q.

In which of the following cases is the function f(x) discontinuous at a?

1. 

2. 

3. 

4. 

Q. $f(2x+3)$

1. $-1\le x\le 1$
2. $-\frac{3}{2}\le x\le -1$
3. $-\frac{3}{2}\le x\le \frac{3}{2}$
4. $-1\le x\le \frac{3}{2}$

Q. For the options given below, the function f(x) = [x] is discontinuous at x =

1. $1$
2. $-0.5$
3. $0.5$
4. $1.5$

Q.

Let $f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$ where $[\cdot ]$ denotes the greatest integer function, then which of the following is true?

1. f(x) has a removable discontinuity at x=1

2. f(x) has a non- removable discontinuity at x=2.

3. f(x) is discontinuous at all positive integers.

4. None of these

Q. For the function $f\left(x\right)=\{\begin{array}{cc}{x}^2-5,& x\le 3\\ \sqrt{x+13},& x>3\end{array}$

1. $1$ only
2. $2$ only
3. Both $1$ and $2$
4. Neither $1$ nor $2$

Q. Match the following lists Let f(x) be any function

List - I	List - II
A) $f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)<0$ then	l) f(x) is increasing at $x=a$
B) $f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)>0$ then	2) f(x) has maximum value at $x=a$
C) $f^{\prime }\left(a\right)\ne 0$ then	3) f(x) has neither maximum nor minimum
D) $f^{\prime }\left(a\right)>0$	4) f(x) has minimum value at $x=a$
	5) f(x) is decreasing at $x=a$

1. A -4, B -2, C -3, D -5.
2. A -2, B -4, C -3, D -1.
3. A -2, B -4, C -3, D -5.
4. A -2, B -4, C -5, D -1.

Q.

$f\left(x\right)=\left[x\right]$ is discontinuous at $x=1$ because

1. Value of function is not defined

2. Value of $f\left(x\right)$ is finite but not equal to the limit

3. Limit doesn't exist

4. Statement wrong, function is continuous

Q.

$li{m}_{x\to 0}\left(\frac{3x+|x|}{7x-5|x|}\right)$=___

1. $\frac{1}{6}$

2. $\frac{3}{7}$

3. 2

4. does not exist

Q.

$f\left(x\right)=\left[x\right]$ is discontinuous at $x=1$ because

1. Value of function is not defined

2. Value of $f\left(x\right)$ is finite but not equal to the limit

3. Limit doesn't exist

4. Statement wrong, function is continuous

Q. The graph of the curve $x^2=3x-y-2$, is

1. between the lines $x=1$ and $x=2$
2. between the lines $x=1$ and $x=\frac{3}{2}$
3. strictly below the line $4y=1$
4. none of these

Q. If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 4\\ -1& 1\end{array}\right]$ then find $AB$

Q. Let $f\left(x\right)=\{\begin{array}{cc}k+x^2,& x<2\\ (x-2{)}^2+3,& x\ge 2\end{array}.$ Then which of the following statements is/are CORRECT?

1. If $f\left(x\right)$ is continuous at $x=2,$ then $k=-1$
2. If $f\left(x\right)$ is differentiable at $x=2,$ then $k=-1$
3. $f\left(x\right)$ is not differentiable at $x=2,$ for any real value of $k$
4. If $k>3,$ then the least value of $f\left(x\right)$ occurs at $x=2$

Q. Plot the graph of ${\sec }^{-1}x$ and write its domain and range.

Q. Let $I$ be any interval disjoint from $[-1,1]$. Prove that the function $f$ given by$f\left(x\right)=x+\frac{1}{x}$ is strictly increasing in interval disjoint from $I$.

Q. Let $f:[-1,3]\to R$ be defined as
$f=\{\begin{array}{ccc}|x|+\left[x\right]& ,& -1\le x<1\\ x+|x|& ,& 1\le x<2\\ x+\left[x\right]& ,& 2\le x\le 3\end{array}$,
where [t] denotes the greatest integer less than or equal to $t$. then, $f$ is discontinuous at:

1. four or more points
2. only one point
3. only three points
4. only two points

Q. If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R$, where $\left[x\right]$ denotes the greatest integer function, then:

1. ${lim}_{x\to 4-}f\left(x\right)$ exists but ${lim}_{x\to 4+}f\left(x\right)$ does not exist
2. ${lim}_{x\to 4+}f\left(x\right)$ exist but ${lim}_{x\to 4-}f\left(x\right)$ does not exist
3. $f$ is continuous at $x=4$
4. Both ${lim}_{x\to 4-}f\left(x\right)$ and ${lim}_{x\to 4+}f\left(x\right)$ exist but are not equal

Q. Find the value of $c$ of Rolle's theorem for $f\left(x\right)=\left|x\right|$ in $[-1,1]$.

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