Question Number 211567 by liuxinnan last updated on 13/Sep/24
$${if}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{g}\left({x}\right)=\mathrm{0} \\ $$$${when}\:{do}\:{not}\:{use}\:{f}\left({x}\right)\:{to}\:\:{replace}\:{g}\left({x}\right)\:\:\: \\ $$
Commented by MrGaster last updated on 13/Sep/24
$$\mathrm{1}.\mathrm{when}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{g}\left({x}\right)=\mathrm{0} \\ $$$$\:\mathrm{cases}\:\mathrm{where}\:\mathrm{you}\:\mathrm{cannot}\:\mathrm{replace}{f}\left({x}\right)\:\mathrm{with}{g}\left({x}\right)\:\:\mathrm{include}\:\mathrm{but}\:\mathrm{are}\:\mathrm{not}\:\mathrm{limitedo} \\ $$$$\mathrm{t}: \\ $$$$\mathrm{1}.\boldsymbol{\mathrm{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{\mathrm{and}}{g}\left({x}\right)\mathrm{are}\:\mathrm{not}\:\mathrm{equivalent}. \\ $$$$\mathrm{infinitesimals} \\ $$$$\mathrm{2}.\mathrm{The}\:\mathrm{specific}\:\mathrm{form}\:\mathrm{of}\:\mathrm{thes} \\ $$$$\mathrm{function}\:\mathrm{is}\:\mathrm{unknown}\:\mathrm{or}\:\mathrm{complex}. \\ $$$$\mathrm{3}.\mathrm{The}\:\mathrm{replacement}\:\mathrm{would}\:\mathrm{affecte} \\ $$$$\mathrm{th}\:\mathrm{results}\:\mathrm{of}\:\mathrm{nonlinearo} \\ $$$$\mathrm{combinatins}. \\ $$$$\mathrm{4}.\mathrm{to}\:\mathrm{indeterminate}\:\mathrm{forms}\:\mathrm{inl} \\ $$$$\mathrm{multipication}\:\mathrm{and}\:\mathrm{divisiont} \\ $$$$\mathrm{operaions}\:\left(\mathrm{such}\:\mathrm{as}\:\mathrm{0}/\mathrm{0}\right). \\ $$
Commented by liuxinnan last updated on 13/Sep/24
$${thanks}\: \\ $$