# Calculate the following limits, if they exist, by using a combination of polar c

glasskerfu 2021-10-22 Answered
Calculate the following limits, if they exist, by using a combination of polar coordinates and de L’Hopital rule.
$\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{\mathrm{arctan}\left({x}^{2}+{y}^{2}\right)}{{x}^{2}+{y}^{2}}$
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## Expert Answer

l1koV
Answered 2021-10-23 Author has 100 answers
changing into polar coordinates
$x=r\mathrm{cos}\theta$
$y=r\mathrm{sin}\theta$
${x}^{2}+{y}^{2}={r}^{2}{\mathrm{cos}}^{2}\theta +{r}^{2}{\mathrm{sin}}^{2}\theta$
$={r}^{2}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)$
$={r}^{2}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta =1\right)$
when $\left(x,y\right)\to \left(0,0\right)$
$r\to 0$
Now $\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{\mathrm{arctan}\left({x}^{2}+{y}^{2}\right)}{{x}^{2}+{y}^{2}}$
$=\underset{r\to 0}{lim}\frac{\mathrm{arctan}\left({r}^{2}\right)}{{r}^{2}}$
$=\underset{r\to 0}{lim}\frac{\frac{1}{1+{\left({r}^{2}\right)}^{2}}\cdot 2r}{2r}$ (applying L'Hopital rule)
$=\underset{r\to 0}{lim}\frac{1}{1+{r}^{4}}$
$=1$
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Jeffrey Jordon
Answered 2022-06-26 Author has 2313 answers

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