Question

Find $\underset{x\to 1}{\lim }f\left(x\right)$, if $f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1,& x\le 1\\ -x^2-1,& x>1\end{array}\right.$

Answer 1

$$\begin{gathered} \underset{x\to 1}{\lim }f\left(x\right)=? \\ f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1,& x\le 1\\ -x^2-1,& x>1\end{array}\right. \\ \mathrm{LHL}: \\ \underset{x\to 1^{-}}{\lim }f\left(x\right) \\ =\underset{x\to 1^{-}}{\lim }\left(x^2-1\right) \\ \mathrm{Let}x=1-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left({\left(1-h\right)}^2-1\right) \\ =0 \\ \mathrm{RHL}: \\ \underset{x\to 1^{+}}{\lim }f\left(x\right) \\ \underset{x\to 1^{+}}{\lim }\left(-x^2-1\right) \\ \mathrm{Let}x=1+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left(-{\left(1+h\right)}^2-1\right) \\ =-2 \\ \mathrm{LHL}\ne \mathrm{RHL} \\ \mathrm{Thus},\underset{x\to 1}{\lim }f\left(x\right)d\mathrm{oes}\mathrm{not}\mathrm{exist}. \end{gathered}$$