Author Topic: Quiz 2 LEC5101  (Read 121 times)

Wang Jingyao

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Quiz 2 LEC5101
« on: October 01, 2020, 07:23:22 PM »
\begin{align}
Find\ the\ limit\ of\ each\ function\ at\ the\ given\ point,\ or\ explain\ &why\ it\ does\ not\ exist. \\
\notag \\
f(z) = (z-2)log|z-2|\ at\ z_o = 2\\

\end{align}

\begin{align}
\lim_{z \to 2}f(z) &= \lim_{z \to 2}(z-2)log|z-2|\ at\ z_o = 2\\
\notag \\
\because z_0 &= 2\\
\notag \\
\therefore z-2 &\to 0\\
\notag \\
Let\ z' &= z-2\\
\notag \\
\therefore z' &\to 0\\
\end{align}

\begin{align}
\therefore \lim_{z' \to 0}f(z) &= \lim_{z' \to 0} z'log|z'|\\
\notag \\
\lim_{z' \to 0}|f(z)| &= \lim_{z' \to 0} |z'log|z'|| \\
\notag \\
&= \lim_{z' \to 0} \dfrac{log|z'|}{\dfrac{1}{|z'|}} \\
\notag \\
Take\ the\ derviative\ &both\ on\ numerator\ and\ denominator\\
&= \lim_{z' \to 0} \dfrac{\dfrac{1}{|z'|}}{-\dfrac{1}{|z'|^2}}\\
\notag \\
&= 0
\end{align}