Question

Evaluate $\underset{x\to 2}{\lim }f\left(x\right)$ (if it exists), where $f\left(x\right)=\left\{\begin{array}{ll}x-\left[x\right],& x<2\\ 4,& x=2\\ 3x-5,& x>2\end{array}.\right.$

Answer 1

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{cc}x-\left[x\right],& x<2\\ 4,& x=2\\ 3x-5,& x>2\end{array}\right. \\ \mathrm{LHL}: \\ \underset{x\to 2^{-}}{\lim }f\left(x\right) \\ =\underset{x\to 2^{-}}{\lim }\left\{x-\left[x\right]\right\} \\ \mathrm{Let}x=2-h,\mathrm{where}h\to 0. \\ \Rightarrow \underset{h\to 0}{\lim }\left[\left(2-h\right)-\left[2-h\right]\right] \\ =2-\left(1\right) \\ =1 \\ \mathrm{RHL}: \\ \underset{x\to 2^{+}}{\lim }f\left(x\right) \\ =\underset{x\to 2^{+}}{\lim }\left(3x-5\right) \\ \mathrm{Let}x=2+h,\mathrm{where}h\to 0. \\ \Rightarrow \underset{h\to 0}{\lim }\left[3\left(2+h\right)-5\right] \\ =6-5 \\ =1 \\ \mathrm{Here},\mathrm{LHL}=\mathrm{RHL}=1 \\ \therefore \underset{x\to 2}{\lim }f\left(x\right)=1 \end{gathered}$$