Find-lim-x-0-1-x-

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

$$\left.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)=−\infty\:\:\:\boldsymbol{{si}}\:\boldsymbol{{x}}\in\right]\mathrm{0};+\infty\left[\right. \\ $$$$\left.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)=+\infty\:\:\boldsymbol{{si}}\:\boldsymbol{{x}}\in\right]−\infty;\mathrm{0}\left[\right. \\ $$

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