Question

Evaluate ${lim}_{x\to 2}f\left(x\right)$ where $f\left(x\right)=\{\begin{array}{c}x-\left[x\right],x<2\\ 4,x=2\\ 3x-5,x>2\end{array}$

Answer 1

$\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{x\to 2^{-}}{lim}x-\left[x\right]=2-1=1$ $\{e.g[1.99]=1\}$

$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}3x-5=6-5=1$

$\Rightarrow \underset{x\to 2}{lim}f\left(x\right)=1$

But as it is not equal to $f\left(2\right)$,

$\therefore$ f is not continuous at $x=2$.