Which one of the following functions is not c | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) We analyze the continuity of each function on the interval $(0, \pi )$: $1$. $f(x) = \cot x = \frac{\cos x}{\sin x}$. This function is discontinuo

Vedclass · Accepted Answer

$l(x) = \begin{cases} x \sin x & 0 < x \le \frac{\pi}{2} \ \frac{\pi}{2} \sin(x + \pi) & \frac{\pi}{2} < x < \pi \end{cases}$

Vedclass · Answer

(D) We analyze the continuity of each function on the interval $(0, \pi )$: $1$. $f(x) = \cot x = \frac{\cos x}{\sin x}$. This function is discontinuous at $x = \pi /2$ is incorrect,it is discontinuous at $x = n\pi$. Within $(0, \pi )$,$\sin x$ is never $0$,so $f(x)$ is continuous. $2$. $g(x) = \int_{0}^{x} t \sin \frac{1}{t} \, dt$. Since the integrand $t \sin(1/t)$ is bounded and continuous on $(0, \pi )$,the integral function $g(x)$ is continuous on $(0, \pi )$. $3$. $h(x)$ is defined piecewise. At $x = 3\pi /4$,the left-hand limit is $1$. The right-hand limit is $2 \sin(\frac{2}{9} \cdot \frac{3\pi}{4}) = 2 \sin(\frac{\pi}{6}) = 2(1/2) = 1$. Since the limits match,$h(x)$ is continuous. $4$. $l(x)$ is defined piecewise. At $x = \pi /2$,the left-hand limit is $\frac{\pi}{2} \sin(\frac{\pi}{2}) = \frac{\pi}{2}$. The right-hand limit is $\frac{\pi}{2} \sin(\frac{\pi}{2} + \pi) = \frac{\pi}{2} \sin(\frac{3\pi}{2}) = \frac{\pi}{2}(-1) = -\frac{\pi}{2}$. Since $\frac{\pi}{2} 
eq -\frac{\pi}{2}$,$l(x)$ is discontinuous at $x = \pi /2$.

Which one of the following functions is not continuous on $(0, \pi )$?

Similar Questions

Number of points of discontinuity of the function $f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\})$ for $x \in [0, 4]$ is (where $[.]$ and $\{.\}$ denote the greatest integer and fractional part functions,respectively).

If the real valued function $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x \sin x} & \text{if } x < 0 \\ p & \text{if } x = 0 \\ \frac{\log(1 + q \sin x)}{x} & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$,then $p + q =$

If $f(x) = \frac{\log_{\sin |x|} \cos^3 x}{\log_{\sin |3x|} \cos^3 (x/2)}$ for $|x| < \frac{\pi}{3}, x \neq 0$ and $f(0) = 4$,then the number of points of discontinuity of $f$ in $\left( -\frac{\pi}{3}, \frac{\pi}{3} \right)$ is:

Consider $f(x) = [x] + \sqrt{\{x\}}$,where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function. Identify the correct statement.

If $f(x) = \begin{cases} x - 1, & x < 0 \\ \frac{1}{4}, & x = 0 \\ x^2, & x > 0 \end{cases}$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module