### Mathematics - Continuity

#### Question - 1

If $f(x)=\frac { 3x+4tanx }{ x }$ is to be defined to make continuous at x=0, then the defined function should be

• A $f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x\neq 0 \\ 7\quad \quad \quad \quad \quad ,x=0 \end{cases}$
• B $f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x\neq 0 \\ 6\quad \quad \quad \quad \quad ,x=0 \end{cases}$
• C $f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x= 0 \\ 7\quad \quad \quad \quad \quad ,x\neq0 \end{cases}$
• D None of the above

#### Question - 2

The value of f(o), so that the function $f(x)=\frac { 2x-sinx^{ -1 } }{ 2x+tan^{ -1 }x }$is continuous at each point in its domain and is equal to

• A $1\over3$
• B $-{1\over3}$
• C $2\over3$
• D $-2\over3$

#### Question - 3

For what value of k, function $f(x)=\begin{cases} \frac { kcosx+4\quad tan\quad x }{ \pi -2x } ,if\quad x\neq \frac { \pi }{ 2 } \\ 3\quad \quad \quad \quad \quad \quad \ ,if\quad x=\frac { \pi }{ 2 } \end{cases}$ is continuous at x=$\pi\over2$

• A 1
• B 3
• C R
• D 6

#### Question - 4

For what value of k, $f(x)=\begin{cases} \frac { { 2 }^{ x+2 }-16 }{ { 4 }^{ x }-16 } ,x\neq 2 \\ k,\quad \quad \quad x=2 \end{cases}$continuous at x=2?

• A 1
• B $3\over2$
• C 2
• D $1\over2$

#### Question - 5

$f(x)=\begin{cases} |x-a|sin\frac { 1 }{ x-a } ,if\quad x\neq 2 \\ 0,\quad \quad \quad \quad \quad \quad if\quad x=a \end{cases}$

• A continuous at x=a
• B discontinuous at x=a
• C discontinuous for all $x \epsilon R$
• D None of the above

#### Question - 6

For what value of k, the function $f(x)=\begin{cases} \frac { \sqrt { 1+kx } -\sqrt { 1-kx } }{ x } ,if-1\le x<0 \\ \frac { 2x+1 }{ x-2 } ,\quad \quad \quad \quad if\quad 0\le x\le 1 \end{cases}$is continuous at x=0?

• A $1\over2$
• B 1
• C $-{3\over2}$
• D $-{1\over2}$

#### Question - 7

$f(x)=\begin{cases} |x|cos\left( \frac { 1 }{ x } \right) ,x\neq 0 \\ 0,\quad \quad \quad \quad x=0 \end{cases}$

• A discontinuous at x=0
• B continuous at x=0
• C does not exist at x=0
• D None of the above

#### Question - 8

$f(x)=\begin{cases} \frac { { x }^{ 2 } }{ 2 } ,\quad \quad \quad \quad \quad if\quad 0\le x\le 1 \\ { 2x }^{ 2 }-3x+\frac { 3 }{ 2 } ,\quad if\quad 1<x\le 2 \end{cases}$

• A discontinuous at x=1
• B discontinuous at x=2
• C continuous at x=1
• D None of the above

#### Question - 9

If $f(x)=\begin{cases} { x }^{ k }sin\left( \frac { 1 }{ x } \right) ,\quad x\neq 0 \\ 0,\quad \quad \quad \quad \quad x=0 \end{cases}$is continuous at x=0, then

• A $k\epsilon (-\infty,0)$
• B $k\epsilon (1,\infty)$
• C $k\epsilon (-1,\infty)$
• D None of these

#### Question - 10

If $f(x)=\frac { \sqrt { 1+sinx } -\sqrt { 1-sinx } }{ x } ,$then the value of f at x=0, so that f is continuous everywhere, is

• A $1\over4$
• B -1
• C 1
• D 2