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




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

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