Question

Find $\underset{x\to 3}{\lim }f\left(x\right)$, where $f\left(x\right)=\left\{\begin{array}{ll}4,& \mathrm{if}x\ge 3\\ x+1,& \mathrm{if}x<3\end{array}\right.$

Answer 1

$$\begin{gathered} \underset{x\to 3}{\lim }f\left(x\right)=? \\ f\left(x\right)=\left\{\begin{array}{ll}4,& x\ge 3\\ x+1,& x<3\end{array}\right. \\ \mathrm{LHL}: \\ \underset{x\to 3^{-}}{\lim }f\left(x\right) \\ =\underset{x\to 3^{-}}{\lim }\left(x+1\right) \\ \mathrm{Let}x=3-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left(3-h+1\right) \\ =4 \\ \mathrm{RHL}: \\ \underset{x\to 3^{+}}{\lim }f\left(x\right) \\ =\underset{x\to 3^{+}}{\lim }\left(4\right) \\ =4 \\ \therefore \mathrm{LHL}=\mathrm{RHL} \\ \underset{x\to 3}{\lim }f\left(x\right)=4 \end{gathered}$$