How do I prove that $\lim_{z\to0}\frac{\Re(z)\Im(z)}{|z|}=0? $ I've tried it using $\varepsilon-\delta$ language, but can't really get anywhere.
Any hints would be greatly appreciated.
Thanks.
How do I prove that $\lim_{z\to0}\frac{\Re(z)\Im(z)}{|z|}=0? $ I've tried it using $\varepsilon-\delta$ language, but can't really get anywhere.
Any hints would be greatly appreciated.
Thanks.
Clearly neither Re(z) nor Im(z) can have magnitude larger than |z|. So:
$\displaystyle{ \lim_{z\to0}\left|\frac{\mathrm{Re}(z)\mathrm{Im}(z)}{|z|}\right| = \lim_{z\to0}\frac{|\mathrm{Re}(z)|\times|\mathrm{Im}(z)|}{|z|} \le \lim_{z\to0}\frac{|z|\times|z|}{|z|} = \lim_{z\to0}|z| = 0 }$
Since the magnitude goes to zero, the quantity goes to zero.
If we rewrite it
$\lim_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2+y^2}}$
does it look more familiar? You can use polar coordinates or Young's inequality.