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Question Number 184239 by mathlove last updated on 04/Jan/23

f(x)= { ((x−2 if x>3)),((3 if x=3)),((−x+4 if 1≤x<3)),((1 if 1<x)) :} faind lim_(x→1) f(x)=? and lim_(x→3) f(x)=?

$${f}\left({x}\right)=\begin{cases}{{x}−\mathrm{2}\:\:\:\:\:\:\:{if}\:\:{x}>\mathrm{3}}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{if}\:\:{x}=\mathrm{3}}\\{−{x}+\mathrm{4}\:\:\:\:{if}\:\:\mathrm{1}\leqslant{x}<\mathrm{3}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{if}\:\:\mathrm{1}<{x}}\end{cases} \\ $$ $${faind}\:\:\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right)=?\:\:{and}\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}{f}\left({x}\right)=? \\ $$

Answered by SEKRET last updated on 04/Jan/23

lim_(x→1^− ) f(x) = 1 lim_(x→1^+ ) f(x)=3 lim_(x→3^− ) f(x) = 1 lim_(x→3^+ ) f(x)=1 lim_(x→3) f(x) = 3

$$\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1}^{−} } \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1}^{+} } \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{3} \\ $$ $$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{3}^{−} } \:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\:=\:\mathrm{1}\:\:\:\:\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{3}^{+} } \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{1} \\ $$ $$\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{3}} \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\:=\:\mathrm{3} \\ $$

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