Question Number 69319 by Joel122 last updated on 22/Sep/19
$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\:\mathrm{2}{y}^{\mathrm{2}} } \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$
Commented by Joel122 last updated on 22/Sep/19
$$\mathrm{I}\:\mathrm{have}\:\mathrm{approached}\:\left(\mathrm{0},\mathrm{0}\right)\:\mathrm{along}\:\:{x}−\mathrm{axis},\:{y}−\mathrm{axis}, \\ $$$${y}={x},\:{y}\:=\:{x}^{\mathrm{2}} ,\:{y}\:=\:\sqrt{{x}}\:\mathrm{and}\:\mathrm{it}\:\mathrm{give}\:\mathrm{same}\:\mathrm{answer}\:+\infty \\ $$$$\mathrm{So},\:\mathrm{why}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}?\:\mathrm{Should}\:\mathrm{I}\: \\ $$$$\mathrm{try}\:\mathrm{using}\:\mathrm{different}\:\mathrm{approach}? \\ $$
Commented by MJS last updated on 22/Sep/19
$$\mathrm{lim}=\pm\infty\:\mathrm{means}\:\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}\:\mathrm{because} \\ $$$$\pm\infty\notin\mathbb{R} \\ $$
Commented by prof Abdo imad last updated on 23/Sep/19
$${x}={rcos}\theta\:{and}\:{y}\:=\frac{{r}}{\:\sqrt{\mathrm{2}}}{sin}\theta\:\:\:\:\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)\Rightarrow{r}\rightarrow\mathrm{0} \\ $$$${and}\:{lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} }\:={lim}_{{r}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{3}}{{r}^{\mathrm{2}} }\:=+\infty \\ $$$${so}\:{the}\:{limit}\:{is}\:{infinite} \\ $$
Commented by Joel122 last updated on 23/Sep/19
$${thank}\:{you}\:{Sir} \\ $$