Question

What will be the value of $li{m}_{z\to 0}\frac{\overline{z}}{z}$?

• A. 0
• B. 1
• C. $\frac{1}{2}$
• D. does not exist

Answer 1

The correct option is D does not exist

$f\left(z\right)=\frac{\overline{z}}{z}=\frac{x-iy}{x+iy}=\frac{x^2-y^2-2ixy}{x^2+y^2}$


$u(x,y)=\frac{x^2-y^2}{x^2+y^2}$

$v(x,y)=\frac{-2xy}{x^2+y^2}$


If we approach (0,0) along y = mx, we get

$li{m}_{x\to 0}u(x,mx)={lim}_{x\to 0}\frac{x^2-m^2{x}^2}{x^2+m^2{x}^2}=li{m}_{x\to 0}\frac{1-m^2}{1+m^2}$

Which depends on m, Hence in $li{m}_{(x,y)\to (0,0)}$ u(x,y) does not exist

Similarly $li{m}_{(x,y)\to (0,0)}$ u(x,y) does not exist

Hence, $li{m}_{z\to 0}\frac{\overline{z}}{z}$ does not exist.