Toronto Math Forum
MAT334-2018F => MAT334--Tests => Quiz-2 => Topic started by: Victor Ivrii on October 05, 2018, 06:15:04 PM
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Find the limit of each function at the given point $z_0$, or explain
why it does not exist.
\begin{equation*}
f(z)=\frac{z^3-8i}{z+2i},\quad z\ne 2i, \qquad\text{at } z_0=2i.
\end{equation*}
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f(z) = z³+(2i)³i/z+2i
= (z+2i)(z²-2iz+(2i)²)/z+2i
=z²-2iz-4
limz->2i f(z)=limz->2i z²-2iz-4
=(2i)²-2i(2i)-4
= -4+4-4
=-4
Practically unreadable despite all insane html "mathematics".
$$\begin{aligned}f(z) &= \frac{z^3+(2i)^3i}{z+2i}\\
&= \frac{(z+2i)(z^2-2iz+(2i)^2)}{z+2i}\\
&=z^2-2iz-4\\
\lim_{z\to2i} f(z)&=\lim_{z\to2i} z^2-2iz-4\\
&=(2i)^2-2i(2i)-4\\
&= -4+4-4\\
&=-4
\end{aligned}$$