Get Answers to all your Questions

header-bg qa

The following functions are continuous on ( 0/\pi )

Option: 1

tan x


Option: 2

\mathrm{ \int_0^x t \sin \frac{1}{t} d t}


Option: 3

\left\{\begin{array}{cc} 1, & 0<x \leq \frac{3 \pi}{4} \\ 3\sin \left(\frac{2}{9} x\right), & \frac{3 \pi}{4}<x<\pi \end{array}\right.


Option: 4

\left\{\begin{array}{cc} x \sin x, & 0<x \leq \frac{\pi}{2} \\ \frac{\pi}{2} \sin (\pi+x), & \frac{\pi}{2}<x<\pi \end{array}\right.


Answers (1)

best_answer

The function  \mathrm{f(x)=\tan x}  is not defined at \mathrm{x=\frac{\pi}{2}} , so f is not continuous on \mathrm{(0, \pi)}
Since the function \mathrm{g(x)=x \sin \frac{1}{x}}  is continuous on \mathrm{(0, \pi)}  and the integral function of a continuous function is continuous, therefore \mathrm{F(x)=\int_0^x t \sin \frac{1}{t} d t} is continuous on \mathrm{(0, \pi)}

\mathrm{\text { For the function } f(x)=\left\{\begin{array}{cc} 1, & 0<x \leq \frac{3 \pi}{4} \\ 2 \sin \left(\frac{2 x}{9}\right), & \frac{3 \pi}{4}<x<\pi \end{array}\right.}

We have,   \mathrm{\lim f(x)=1}
\mathrm{ x \rightarrow \frac{3 \pi}{4} }
and,
\mathrm{ \lim _{x \rightarrow \frac{3 \pi}{4}} f(x)=\lim _{x \rightarrow \frac{3 \pi}{4}} 2 \sin \left(\frac{2 x}{9}\right)=2 \sin \frac{\pi}{6}=1 \text {. } }
So, f(x) is continuous at   \mathrm{ x=\frac{3 \pi}{4} }

Evidently f(x) is continuous at all other points.
For the function \mathrm{ f(x)=\frac{\pi}{2} \sin (\pi+x) \cdot f\left(\frac{\pi}{2}\right)=\frac{\pi}{2} }
\mathrm{ \lim _{x \rightarrow \frac{\pi}{2}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{\pi}{2}-h\right)=\lim _{h \rightarrow 0} \frac{\pi}{2} \sin \left(\frac{3 \pi}{2}-h\right)=\frac{\pi}{2} } 

and, \mathrm{ \lim _{x \rightarrow \frac{\pi \cdot}{2}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{\pi}{2}+h\right)=\lim _{h \rightarrow 0} \frac{\pi}{2} \sin \left(\frac{3 \pi}{2}+h\right)=\frac{\pi}{2} }
So, f(x) is not continuous at   \mathrm{ x=\frac{\pi}{2} }.

Posted by

Ritika Harsh

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE