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Question Number 69319 by Joel122 last updated on 22/Sep/19

Show that   lim_((x,y)→(0,0))  (3/(x^2  + 2y^2 ))  does not exist

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\:\mathrm{2}{y}^{\mathrm{2}} } \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$

Commented by Joel122 last updated on 22/Sep/19

I have approached (0,0) along  x−axis, y−axis,  y=x, y = x^2 , y = (√x) and it give same answer +∞  So, why the limit doesn′t exist? Should I   try using different approach?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{approached}\:\left(\mathrm{0},\mathrm{0}\right)\:\mathrm{along}\:\:{x}−\mathrm{axis},\:{y}−\mathrm{axis}, \\ $$$${y}={x},\:{y}\:=\:{x}^{\mathrm{2}} ,\:{y}\:=\:\sqrt{{x}}\:\mathrm{and}\:\mathrm{it}\:\mathrm{give}\:\mathrm{same}\:\mathrm{answer}\:+\infty \\ $$$$\mathrm{So},\:\mathrm{why}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}?\:\mathrm{Should}\:\mathrm{I}\: \\ $$$$\mathrm{try}\:\mathrm{using}\:\mathrm{different}\:\mathrm{approach}? \\ $$

Commented by MJS last updated on 22/Sep/19

lim=±∞ means it doesn′t exist because  ±∞∉R

$$\mathrm{lim}=\pm\infty\:\mathrm{means}\:\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}\:\mathrm{because} \\ $$$$\pm\infty\notin\mathbb{R} \\ $$

Commented by prof Abdo imad last updated on 23/Sep/19

x=rcosθ and y =(r/(√2))sinθ    (x,y)→(0,0)⇒r→0  and lim_((x,y)→(0,0))    (3/(x^2  +2y^2 )) =lim_(r→0)    (3/r^2 ) =+∞  so the limit is infinite

$${x}={rcos}\theta\:{and}\:{y}\:=\frac{{r}}{\sqrt{\mathrm{2}}}{sin}\theta\:\:\:\:\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)\Rightarrow{r}\rightarrow\mathrm{0} \\ $$$${and}\:{lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} }\:={lim}_{{r}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{3}}{{r}^{\mathrm{2}} }\:=+\infty \\ $$$${so}\:{the}\:{limit}\:{is}\:{infinite} \\ $$

Commented by Joel122 last updated on 23/Sep/19

thank you Sir

$${thank}\:{you}\:{Sir} \\ $$

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