 # Find the solution of limit \lim_{(x,y)->0,0} sqrt((x^2+y^2)/(x^2+y^2)) by using the polar coordinates system necessaryh 2021-01-17 Answered

Find the solution of limit $\underset{\left(x,y\right)\to 0,0}{lim}\frac{\sqrt{{x}^{2}+{y}^{2}}}{{x}^{2}+{y}^{2}}$ by using the polar coordinates system.

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$\underset{\left(x,y\right)\to 0,0}{lim}\left(\sqrt{{x}^{2}+{y}^{2}}\left({x}^{2}+{y}^{2}\right)\right)=\underset{\left(x,y\right)\to 0,0}{lim}f\left(x,y\right)$,
where we have
$f\left(r,0\right)=\frac{r}{{r}^{2}\left({\mathrm{cos}}^{2}0+{\mathrm{sin}}^{2}0\right)}=\frac{r}{{r}^{2}}$
Now
$\underset{r\to 0,0\to 0}{lim}f\left(r,0\right)=\underset{r\to 0}{lim}\frac{r}{{r}^{2}}=\mathrm{\infty }$
Therefore
$\underset{\left(x,y\right)\to 0,0}{lim}\frac{{\sqrt{x}}^{2}+{y}^{2}}{{x}^{2}+{y}^{2}}=\mathrm{\infty }$