If f(x)= c 1 ; when 0< x 3 4 2 2 9 x ; when 3 4 < x< ., then - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{c}1 ; \text { when } 0< x \leq \frac{3 \pi}{4} \\ 2 \sin \frac{2}{9} x ; \text { when } \frac{3 \pi}{4}< x<\pi\end{array}\right.$, then

A
$f (x)$ is continuous at $x=0$
B
$f (x)$ is continuous at $x=\pi$
✓
$f(x)$ is continuous at $x=\frac{3 \pi}{4}$
D
$f (x)$ is discontinuous at $x=\frac{3 \pi}{4}$

✓

Answer

Correct option: C.

$f(x)$ is continuous at $x=\frac{3 \pi}{4}$

(C)
Here, $f \left(\frac{3 \pi}{4}\right)=1$ and $\lim _{x \rightarrow \frac{3 \pi^{-}}{4}} f (x)=1$
$\lim _{x \rightarrow \frac{3 \pi^{+}}{4}} f (x)=\lim _{ h \rightarrow 0} 2 \sin \frac{2}{9}\left(\frac{3 \pi}{4}+ h \right)$
$=2 \sin \frac{\pi}{6}=1$
$\therefore \quad f (x)$ is continuous at $x-\frac{3 \pi}{4}$.

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