Question

Discuss the continuity of the following functions :

(c) f(x) = sin x cos x

Answer 1

Here, f(x) =sin x cos x

$\frac{1}{2}\times 2sinxcosx=\frac{1}{2}sin2x$

At x=a, where $aϵR$

LHL = $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{-}}{lim}\frac{1}{2}sin2x=\underset{x\to 0^{-}}{lim}\frac{1}{2}sin2(a-h)$

=$\underset{x\to a^{-}}{lim}\frac{1}{2}[sin2acos2h-cos2asin2h]$

=$\frac{1}{2}[sin2acos0-cos2asin0]=\frac{1}{2}sin2a$ [Use formula sin(A+B)]

LHL = $\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}\frac{1}{2}sin2x=\underset{h\to 0}{lim}\frac{1}{2}sin2(a+h)$

=$\underset{h\to 0}{lim}\frac{1}{2}[sin2acos2h-cos2asin2h]$[Use formula sin(A-B)]

=$\underset{h\to 0}{lim}\frac{1}{2}\left[sin\right(2a+2h\left)\right]\frac{1}{2}sin2a$

Also, f(a) =$\frac{1}{2}$~sin~2a

$\therefore$ LHL=RHL=f(a). Hence, f(x) is continuous at all points.