Tag: continuity

Quizzes (2) Questions (26)

Questions Related to continuity

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

Number of points of discontinuity of $f\left( x \right) = \left[ {2{x^3} - 5} \right]$ in $\left[ {1,2} \right)$ is where $\left[ x \right]$ denotes greatest integer function are

  1. $14$
  2. $13$
  3. $10$
  4. $8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The function f(x) = [2x^3 - 5] is discontinuous where the expression inside the bracket is an integer. For x in [1, 2), 2x^3 - 5 ranges from 2(1)^3 - 5 = -3 to 2(2)^3 - 5 = 11. The integer values are -3, -2, ..., 10. There are 14 such values.

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

$f(x)=\displaystyle\lim _{n\rightarrow \infty}\dfrac{(x-1)^{2n}-1}{(x-1)^{2n}+1}$ is discontinuous at

  1. $x=0$ only
  2. $x=2$ only
  3. $x=0$ and $2$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The limit exists as 1 if |x-1| > 1 (i.e., x > 2 or x < 0) and as -1 if |x-1| < 1 (i.e., 0 < x < 2). At x=0 and x=2, the limit does not exist because the left and right limits differ, causing discontinuity.

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

The sum of all values of $x$ for which $f(x)=[3\sin x]$ is discontinous in $[0,\ 2\pi]$ is (where [.] represents greatest integers function)

  1. $\dfrac {21\pi}{2}$
  2. $13\ \pi$
  3. $11\ \pi$
  4. $\dfrac {23\pi}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

f(x) = [3 sin x] is discontinuous when 3 sin x is an integer. In [0, 2pi], 3 sin x takes integer values -3, -2, -1, 0, 1, 2, 3. Solving 3 sin x = k for k in {-2, -1, 0, 1, 2} yields multiple points. Summing these points leads to 13pi.

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

Consider the function defined on $[0,\ 1]\rightarrow R,\ f(x)=\dfrac {\sin x-x\cos x}{x^{2}}$ if $x\neq 0$ and $f(0)=0$ then the function of $f(x)$. 

  1. Has a removable discontinuity at $x=0$
  2. Has a removable finite discontinuity at $x=0$
  3. Has a non removable infinite discontinuity at $x=0$
  4. Is continuous at $x=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The limit of (sin x - x cos x) / x^2 as x approaches 0 is 0. Since the function is defined as 0 at x=0, and the limit exists and equals the function value, the discontinuity is removable.

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

The function $f(x)={ sin }^{ -1 }(cosx)$ is :

  1. Discontinuous at x = 0

  2. Continuous at x = 0

  3. Differentiable at x = 0

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$f(x)=\sin^{-1} (\cos x)$

LHL :
$lim _{x\rightarrow 0^-} \sin^{-1}(\cos x)=lim _{h\rightarrow 0} \sin^{-1}(\cos (0-h))$

$lim _{h\rightarrow 0} \sin^{-1} (\cos (-h))=lim _{h\rightarrow 0} \sin^{-1}(\cos 0)=\dfrac{\pi}{2}$

RHL:
$lim _{x\rightarrow 0^{+}} \sin^{-1}(\cos x)$
$=lim _{h\rightarrow 0} \sin^{-1} \cos (0+h)$
$=\sin^{-1} \cos 0$
$=\dfrac{\pi}{2}$

Thus, $LHL=RHL=f(0)=\dfrac{\pi}{2}$

RHD :
$lim _{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}=\dfrac{sin^{-1}(\cos h)-1}{h}$

$lim _{h\rightarrow 0} \dfrac{\sin^{-1}(\cos h)-1}{h}=\dfrac{1-\sin h}{\sqrt{1-\cos^2 h}}=\dfrac{-\sin h}{\sin h}=-1$

LHD :
$lim _{h\rightarrow 0} \dfrac{f(x-h)-f(x)}{-h}=lim _{h\rightarrow 0}\dfrac{\sin^{-1}(\cos -h)-1}{-h}$

$lim _{h\rightarrow 0} \dfrac{\sin^{-1}(\cos h)-1}{-h}=\dfrac{-\sin h}{-\sin h}=1$

$LHD \neq RHD$
Thus, function is not differentiable at $x=0$.

Multiple choice mathematics and statistics continuity discontinuity and its types types of discontinuity differencial calculus - limits and continuity

If $f\left( x \right) =\begin{cases} -1,if\ x<0\ \ 0,if\ x=0\ \ 1,if\ x>0\ \end{cases}$ and $g\left(x\right)=\sin x +\cos x$, then point discontinuity of $(fog)(x)$ in $(0,2\pi)$ are 

  1. $\dfrac{\pi}{4},\dfrac{5\pi}{4}$
  2. $\dfrac{\pi}{4},\dfrac{3\pi}{4}$
  3. $\dfrac{\pi}{4},\dfrac{7\pi}{4}$
  4. $\dfrac{3\pi}{4},\dfrac{7\pi}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer