Question Number 73570 by Rio Michael last updated on 13/Nov/19
$${f}\::\:{x}\:\rightarrow\:\begin{cases}{\mathrm{1}\:+\:{x},\:{if}\:{x}<\mathrm{1}}\\{\mathrm{2}{x}−\mathrm{1},{if}\:{x}>\mathrm{1}}\end{cases} \\ $$$${investigate}\:{the}\:{existence}\:{and}\:{non}\:{existence}\:{of}\:{the} \\ $$$${limit}\:{of}\:{f}\:{at}\:{the}\:{point}\:{x}\:=\mathrm{1} \\ $$
Commented by kaivan.ahmadi last updated on 13/Nov/19
$${lim}_{{x}\rightarrow\mathrm{1}^{−} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{1}^{−} } \left(\mathrm{1}+{x}\right)=\mathrm{2} \\ $$$${lim}_{{x}\rightarrow\mathrm{1}^{+} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{1}^{+} } \left(\mathrm{2}{x}−\mathrm{1}\right)=\mathrm{1} \\ $$$$\Rightarrow{lim}_{{x}\rightarrow\mathrm{1}^{−} } {f}\left({x}\right)\neq{lim}_{{x}\rightarrow\mathrm{1}^{+} } {f}\left({x}\right)\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)\:\:{is}\:{not}\:{exist}. \\ $$