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

Question

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

Answer 1

lim_{(x, y) \to 0, 0} (\sqrt{x^{2} + y^{2}} (x^{2} + y^{2})) = lim_{(x, y) \to 0, 0} f (x, y)

,
where

f (x, y) = (\sqrt{x^{2} + y^{2}}) . Using polar form x = r \cos 0, y = r \sin 0

we have

f (r, 0) = \frac{r}{r^{2} (\cos^{2} 0 + \sin^{2} 0)} = \frac{r}{r^{2}}

Now

lim_{r \to 0, 0 \to 0} f (r, 0) = lim_{r \to 0} \frac{r}{r^{2}} = \infty

Therefore

lim_{(x, y) \to 0, 0} \frac{{\sqrt{x}}^{2} + y^{2}}{x^{2} + y^{2}} = \infty

Expert Answer