Question Number 205256 by hardmath last updated on 13/Mar/24
$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:? \\ $$
Commented by mr W last updated on 13/Mar/24
$${your}\:{earnest}? \\ $$
Commented by hardmath last updated on 13/Mar/24
$$ \\ $$Dear profesdor, the example says so
Answered by Frix last updated on 13/Mar/24
$$\mathrm{Let}\:{t}=\frac{\mathrm{1}}{{x}}\:\Leftrightarrow\:{x}=\frac{\mathrm{1}}{{t}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}}\:=\underset{{t}\rightarrow−\infty} {\mathrm{lim}}\:{t}\:=−\infty \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}}\:=\underset{{t}\rightarrow+\infty} {\mathrm{lim}}\:{t}\:=+\infty \\ $$$$\Rightarrow \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$
Answered by lepuissantcedricjunior last updated on 13/Mar/24
![lim_(x→0) ((1/x))=−∞ si x∈]0;+∞[ lim_(x→0) ((1/x))=+∞ si x∈]−∞;0[](Q205268.png)
$$\left.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)=−\infty\:\:\:\boldsymbol{{si}}\:\boldsymbol{{x}}\in\right]\mathrm{0};+\infty\left[\right. \\ $$$$\left.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)=+\infty\:\:\boldsymbol{{si}}\:\boldsymbol{{x}}\in\right]−\infty;\mathrm{0}\left[\right. \\ $$