Question Number 211570 by liuxinnan last updated on 13/Sep/24
$$\left\{{x}_{{n}} \right\}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$${prove}\:{when}\:{a}<\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\mathrm{0} \\ $$$${when}\:{a}>\mathrm{1}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\infty \\ $$$${and}\:{when}\:{a}=\mathrm{1}\:{what}\:{happen}\:{about}\:{x}_{{n}} \\ $$
Commented by MrGaster last updated on 13/Sep/24
$$\mathrm{Case}\:\mathrm{1}:{a}>\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$$\mathrm{2}.\mathrm{Choose}\::\epsilon={b}−{a}>\mathrm{0where}\: \\ $$$${a}<{b}<\mathrm{1} \\ $$$$\mathrm{3}.{a}−\epsilon<{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} {a}+\epsilon \\ $$$$\mathrm{4}.\mathrm{2}{a}−{b}<{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} <{b} \\ $$$$\mathrm{5}.{x}_{{n}} <{b}^{{n}} \\ $$$$\mathrm{6}.\mathrm{A}{s}\:{n}\rightarrow\infty,{b}^{{n}} \rightarrow\mathrm{0} \\ $$$$\mathrm{7}.\mathrm{By}\:\mathrm{squeeze}\:\mathrm{theorem},\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}_{{n}} =\mathrm{0} \\ $$$$\mathrm{Case}\:\mathrm{2}:{a}>\mathrm{1} \\ $$$$\mathrm{1}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$$\mathrm{2}.\mathrm{Choose}\:\epsilon={a}−{c}>\mathrm{0}\:\mathrm{where} \\ $$$$\mathrm{1}<{c}<{a} \\ $$$$\mathrm{3}.{a}−\epsilon<{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} <{a}+\epsilon \\ $$$$\mathrm{4}.{c}<{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} <\mathrm{2}{a}−{c} \\ $$$$\mathrm{5}.{x}_{{n}} >{c}^{{n}} \\ $$$$\mathrm{6}.{As}\:{n}\rightarrow\infty,{c}^{{n}} \rightarrow\infty \\ $$$$\mathrm{7}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}_{{n}} =\infty \\ $$$$\mathrm{Case}\:\mathrm{3}:{a}=\mathrm{1} \\ $$$$\mathrm{1}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a}=\mathrm{1} \\ $$$$\mathrm{2}.\mathrm{The}\:\mathrm{behavior}\:\mathrm{of}\:\left\{\boldsymbol{{x}}_{\boldsymbol{{n}}} \right\}\mathrm{as}\:{x}\rightarrow\mathrm{0}\:\mathrm{is} \\ $$$$\mathrm{not}\:\mathrm{determined}. \\ $$$$\mathrm{3}.\mathrm{Without}\:\mathrm{additional}\: \\ $$$$\mathrm{informationthe}\:\mathrm{limit}\:\mathrm{of}\:{x}_{{n}} \mathrm{cannot}\:\mathrm{be}\:\mathrm{determined}.. \\ $$
Commented by liuxinnan last updated on 13/Sep/24
$${nice} \\ $$