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f-x-1-x-if-x-lt-1-2x-1-if-x-gt-1-investigate-the-existence-and-non-existence-of-the-limit-of-f-at-the-point-x-1-




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

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