Question Number 221392 by Davidtim last updated on 02/Jun/25
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}=? \\ $$$$\left.\mathrm{1}\right)\:\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{3} \\ $$$$\left.\mathrm{3}\right)\:{Does}\:{not}\:{exist} \\ $$$$\left.\mathrm{4}\right)\:{Undefined} \\ $$
Answered by Frix last updated on 02/Jun/25
![lim_(x→3) (√(x−3)) =^([x=t+3]) lim_(t→0) (√t) lim_(t→0^− ) (√t) does not exist lim_(t→0^+ ) (√t) =0 lim_(t→0^− ) (√t) ≠lim_(t→0^+ ) (√t) ⇒ lim_(t→0) (√t) does not exist ⇒ lim_(x→3) (√(x−3)) does not exist](https://www.tinkutara.com/question/Q221395.png)
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\sqrt{{x}−\mathrm{3}}\:\overset{\left[{x}={t}+\mathrm{3}\right]} {=}\:\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{{t}} \\ $$$$\underset{{t}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\sqrt{{t}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$$$\underset{{t}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\sqrt{{t}}\:=\mathrm{0} \\ $$$$\underset{{t}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\sqrt{{t}}\:\neq\underset{{t}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\sqrt{{t}}\:\Rightarrow\:\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{{t}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$$$\Rightarrow \\ $$$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\sqrt{{x}−\mathrm{3}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$
Commented by Davidtim last updated on 02/Jun/25
$${It}\:{is}\:{undefined},\:{because}; \\ $$$$\underset{{x}\rightarrow\mathrm{3}^{+} } {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}{DNE}\:{in}\:{real}\:{numbers} \\ $$$$\Rightarrow\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}={Undefined} \\ $$
Commented by Davidtim last updated on 02/Jun/25
$${It}\:{is}\:{undefined},\:{because}; \\ $$$$\underset{{x}\rightarrow\mathrm{3}^{+} } {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}{DNE}\:{in}\:{real}\:{numbers} \\ $$$$\Rightarrow\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}={Undefined} \\ $$
Commented by Frix last updated on 02/Jun/25
$$\mathrm{If}\:\mathrm{it}'\mathrm{s}\:\mathrm{undefined}\:\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}. \\ $$$$\mathrm{This}\:\mathrm{might}\:\mathrm{be}\:\mathrm{a}\:\mathrm{problem}\:\mathrm{of}\:\mathrm{translating}\:\mathrm{from} \\ $$$$\mathrm{different}\:\mathrm{languages}\:\mathrm{to}\:\mathrm{English}. \\ $$