Question

For a complex number $z$, ${lim}_{z\to i}\frac{z^2+1}{z^3+2z-i(z^2+2)}$ is

• A. $-2i$
• B. $-i$
• C. $i$
• D. $2i$

Answer 1

The correct option is D $2i$

${lim}_{z\to i}\frac{z^2+1}{z^3+2z-i(z^2+2)}$
This is $\frac{0}{0}$ from, so on differentiaitng both numerator and denominator
${lim}_{z\to i}\frac{2z}{3z^2+2-2zi}$
$=\frac{2i}{3\left(i^2\right)+2-2\left(i\right)i}=\frac{2i}{-3+2+2}=2i$