Question

Examine the continuity of the function
$f\left(x\right)=\{\begin{array}{c}(\sin x-\cos x)\text{for}x\ne 0\\ -1\text{for}x=0\end{array}$
at the point $x=0$

Answer 1

L.H.L $=\underset{x\to 0^{-}}{lim}f\left(x\right)$
$\underset{x\to 0^{-}}{lim}(\sin x-\cos x)$
$=0-1$
$=-1$
R.H.L $={lim}_{x\to 0^{+}}f\left(x\right)$
$={lim}_{x\to 0^{+}}(\sin x-\cos x)=-1$
${lim}_{x\to 0^{}}f\left(x\right)=-1$ ....(Given)
L.H.L $=$ R.H.L $={lim}_{x\to 0^{}}f\left(x\right)$