Continuity of a Function

Q.

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{\frac{1}{x}}-e+\frac{1}{2}ex}{x^2}$ is ---------

1. 

2. 

3. 

4. 

Q.

Find the domain and range of the real function f defined by $f\left(x\right)=\sqrt{x-1}$.

Q.

If $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ be in H.P., then a, b, c are in

1. A.P

2. H.P

3. G.P

4. None of these

Q.

A mapping $f:n⇢N$, where $N$ is the set of natural numbers is defined as

$f\left(n\right)=\left\{\begin{array}{l}{n}^2,fornodd\\ 2n+1,forneven\end{array}\right.$

for $n$ belongs to $N$. Then $f$is

1. surjective but not injective

2. injective but not surjective

3. bijective

4. neither injective nor surjective

Q.

Find the domain and range of the function given by $f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}$, where [x] is greatest integer function.

Q.

Consider the function f : R → R defined by $f\left(x\right)=\left\{\begin{array}{l}\{2-\sin (1/x\left)\right\}\left|x\right|x\ne 0\\ 0x=0\end{array}\right.$

Then f is

1. monotonic on $(0,\infty )$ only

2. not monotonic on $(-\infty ,0)$ and$(0,\infty )$

3. monotonic on $(-\infty ,0)$ only

4. monotonic on $(-\infty ,0)\cup (0,\infty )$

Q.

Let $f\left(x\right)=[x{]}^2+\sqrt{\left\{x\right\}}$, where [x] is greatest integer function and {x} is the fractional part function, then

the function f(x) is discontinuous at.

1. All integer points

2. x = 0

3. x = 1

4. All integer points except x = 1

Q. Let $f\left(x\right)=[cosx+sinx],0<x<2\pi ,$ where [x] denotes the greatest integer less than of equal to x. The number of points of discontinuity of f(x) is

1. 5
2. 6
3. 4
4. 3

Q. If $R$ is the set of real numbers and $Q$ is the set of rational numbers, then what is $R-Q$?

Q.

Let $f$ be a real-valued function, defined on $R-\left\{-1,1\right\}$ and given by $f\left(x\right)=3{\log }_e\left|\frac{\left(x-1\right)}{\left(x+1\right)}\right|-\frac{2}{\left(x-1\right)}$. Then in which of the following intervals, function $f\left(x\right)$ is increasing?

1. $\left(-\infty ,-1\right)\cup \left([1/2,\infty )-\left\{1\right\}\right)$

2. $(-1,1/2]$

3. $\left(-\infty ,\infty \right)-\left\{-1,1\right\}$

4. $(-\infty ,1/2]-\left\{-1\right\}$

Q. If f(x) be a continuous function defined for $1\le x\le 3.f\left(x\right)ϵQ\forall xϵ[1,3]andf\left(2\right)=10$ (Where Q is a set of all rational numbers). Then, f(1.8) is

1. 1
2. 5
3. 10
4. 20

Q.

$\underset{x\to 0}{lim}{x}^5\left[\frac{1}{x^3}\right]$ where [.] denotes the greatest integer function.

1. 1

2. 

3. 0

4. -1

Q. The value of $a$ for which the function $f\left(x\right)=a\sin x+\frac{1}{3}\sin 3x$ has an extremum at $x=\frac{\pi }{3}$ is

Q. Let f(x) = $\{\begin{array}{cc}\frac{x^3+x^2-16x+20}{(x-2{)}^2},& ifx\ne 2\\ k,& ifx=2\end{array}.$ if f(x) be continuous for all x, then k =

1. 7
2. -7
3. $\pm 7$
4. None of these

Q. If $f\left(x\right)=\{\begin{array}{cc}xsinx,& when0<x\le \frac{\pi }{2}\\ \frac{\pi }{2}sin(\pi +x),& when\frac{\pi }{2}<x<\pi \end{array},then$

1. f(x) is discontinuous at x = π / 2
2. f(x) is continuous at x = π / 2
3. f(x) is continuous at x = 0
4. None of these

Q. Given $f\left(x\right)=\{\begin{array}{ccc}\frac{\sin x}{2x}& ,& x<0\\ a& ,& x=0\\ \frac{1-\cos 2x}{b{x}^2}& ,& x>0\end{array}$

If f(x) is continuous at x = 0, then which of the following is/are correct?

दिया है $f\left(x\right)=\{\begin{array}{ccc}\frac{\sin x}{2x}& ,& x<0\\ a& ,& x=0\\ \frac{1-\cos 2x}{b{x}^2}& ,& x>0\end{array}$

यदि x = 0 पर, f(x) सतत है, तब निम्नलिखित में से कौनसा/कौनसे विकल्प सही है/हैं?

1. a = 1
   
2. $b=\frac{1}{4}$
   
3. $a=\frac{1}{2}$
   
4. b = 4
   

Q. The function $f\left(x\right)=\frac{log(1+ax)-log(1-bx)}{x}$ is not defined at x-0 . The value which should be assigned to f at x = 0 so that it is continuos at x=0 , is

1. a-b
2. loga+logb
3. log a-log b
4. a+b

Q.

$\underset{x\to 0}{lim}\frac{e^x-1}{x}=$

1. -1

2. 1

3. 

4. e

Q. The function f(x) defined as f(x) = $\sqrt{(x-4{)}^2}$

1. not continuous at x = 4
2. continuous but not differentiable $\forall xϵR$
3. continuous and differentiable $\forall xϵR$
4. continuous only for $x\ge 4$

Q.

$\text{The values of constants a and b so that}\underset{x\to \infty }{lim}(\frac{x^2+1}{x+1}-ax-b)=\frac{1}{2},are$

1. 

2. 

3. 

4. 

Q.

Figure shows a relation between set P and Q. Write the relation R

1. R = {(9, 3), (9, -3), (4, 2), (4, -2), (36, 6), (36, -6)}

2. R = {(9, 1), (9, 2), (9, 3), (9, 4), (9, 5), (9, 6)}

3. R = {(9, 3), (9, -3), (4, 2), (4, -2), (25, 5), (25, -5), (36, 6), (36, -6)}

4. R = {(1, 7), (2, 4), (-2, 4), (3, 9), (-4, 16), (4, 16)}

Q.

If a, b, c are three distinvt positive real numbers which are in H.P., then $\frac{3a+2b}{2a-b}+\frac{3c+2b}{2c-b}$ is

1. 

2. 

3. 

4. 

Q.

The value of ${lim}_{x\to \infty }\frac{x^5}{5^x}$ is

1. 1

2. -1

3. 0

4. 1/2

Q. The value of k for which $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x-x\cos x}{x^3\cos x},& x\ne 0\\ k,& x=0\end{array}$ is continuous at x = 0, is

k के किस मान के लिए, $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x-x\cos x}{x^3\cos x},& x\ne 0\\ k,& x=0\end{array}$, x = 0 पर सतत है?

1. $\frac{1}{2}$
   
2. $\frac{-1}{6}$
   
3. $\frac{1}{6}$
   
4. $\frac{1}{3}$
   

Q.

If A + B + C = $\pi$. Then

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1.sin2A + sin2B + sin2C}& \text{A.cos}\frac{A}{2}\text{cos}\frac{B}{2}\text{cos}\frac{C}{2}\\ \text{2.sinA + sinB + sinC}& \text{B.1+4sin}\frac{A}{2}\text{sin}\frac{B}{2}\text{sin}\frac{C}{2}\\ \text{3.cos2A + cos2B + cos2C}& \text{C.4sinA sinB sinC}\\ \text{4.cosA + cosB + cosC}& \text{D.-1-4cosA + cosB + cosC}\end{array}$

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

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

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

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

Q.

If $S_n$ = $\sum _{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n$ = $\sum _{r=0}^n\frac{r}{{}^n{C}_r}$, then $\frac{t_n}{s_n}$, when n =100 equals to

__

Q.

For each of the following statements, determine whether an inclusive 'or' or exclusive 'or' is used. Give reasons for your answer.

(i) For identification you need a passport or an Ahar Card.

(ii) the school is closed if it is a holiday or a Sunday.

(iii) $\sqrt{3}$ is a rational number or an irrational number.

(iv) Two lines intersect at a point or are parallel.

(v) Students can take Sanskrit or French as their third language.

Q. Given that $f\left(x\right)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ up to infinite terms.

The derivative of $f\left(x\right)$ is .

1. $f\left(x\right)$
2. $2f\left(x\right)$
3. $\left[f\right(x){]}^2$
4. $\frac{f\left(x\right)}{x!}$

Q.

Find the value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{x}$

1. limit does not exist

2. $0$

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

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

Q. Let f(x) = $\{\begin{array}{cc}\frac{x^3+x^2-16x+20}{(x-2{)}^2},& ifx\ne 2\\ k,& ifx=2\end{array}.$ if f(x) be continuous for all x, then k =

1. None of these
2. -7
3. 7
4.