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
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